Wayne Rooney scored a last-minute penalty as a poor Manchester United edged League One Sheffield United to reach the FA Cup fourth round.
The hosts, 47 places above their opponents in the football pyramid, had only two shots on target throughout.
Substitute Memphis Depay curled just wide in the second half, but it looked like the visitors would earn a replay.
However, after Memphis was fouled inside the area, Rooney stepped up to score and send the hosts through.
Read about how the game unfolded
Reaction to Saturday's FA Cup third-round games
Just two weeks ago Louis van Gaal's future as Manchester United manager appeared to be in huge doubt.
The Red Devils had failed to win in seven games but, after surviving a 2-0 defeat at Stoke on Boxing Day, Van Gaal has overseen an improvement in both results and, in patches, performances.
After a combined 31 shots on goal in their last two games against Chelsea and Swansea, Manchester United seemed a livelier prospect, despite their plentiful possession still having no end product.
And, at Old Trafford on Saturday, Memphis fizzed an effort just wide in the second half, only for the home fans to burst into ironic cheers, rather than roar encouragement.
A drab draw against a League One side would likely have seen Van Gaal come under criticism once again, but Rooney's strike could give him some breathing space.
Back in 1990, Mark Robins scored a goal in the FA Cup for Manchester United that is said to have saved Sir Alex Ferguson from the sack.
Van Gaal may have just had his 'Robins moment'.
Sheffield United looked to have secured a lucrative replay at Bramall Lane but, as the game entered the 90th minute, Memphis broke into the area and fell as he was challenged by midfielder Dean Hammond.
His acrobatic reaction suggested the hosts may have got lucky when referee Jon Moss pointed to the spot, but replays showed there was limited contact.
"I thought it was a penalty but it was an absolute shocking performance from Manchester United," BSports Radio 5 live pundit Robbie Savage said.
"For me, Van Gaal, I'd get rid of him."
Two years ago Sheffield United reached the semi-finals of the FA Cup and had forged a reputation as seasoned giant killers.
In recent years, they have beaten Aston Villa, West Ham and Southampton in cup competitions and, given the sense of unrest at Old Trafford, they will have fancied their chances of claiming a higher division scalp.
They had the game's first shot on target, which came in the 67th minute after Chris Basham fired straight at David De Gea, but retreated into their own half as they looked to keep Manchester United out and take the tie back to South Yorkshire.
That perhaps proved their undoing as the hosts pressed in the closing stages, leading to the moment when Memphis was brought down inside the box.
A little more conviction, and the result could have been different.
"There just seems to be an acceptance of bad performances," former Manchester United midfielder Paul Scholes told BT Sport.
"Every time you come to Old Trafford you see negative football."
from Scholes
Manchester United manager Louis van Gaal: "You have to see also how the opponent is playing. Everybody shall give applause to Sheffield United. They were very organised but it is much more difficult to attack than to defend.
"Again a clean sheet is also the benefit of our organisation. I'm very disappointed that we are not creating too many chances, that we have to improve."
Sheffield United boss Nigel Adkins on BT Sport: "The players have worked extremely hard against superior opposition.
"The grit, desire and character gave the supporters something to cheer about. The supporters have got right behind the players and there is a togetherness at the football club."
Manchester United travel to Newcastle in the Premier League on Tuesday, while Sheffield United are back in League One action as they face 2013 FA Cup winners Wigan.
The draw for the FA Cup fourth round takes place on Monday and is live on The One Show on BSports One from 19:00 GMT.
Full Time Match ends, Manchester United 1, Sheffield United 0.
90:00 +7:14 Full timeFull Time Second Half ends, Manchester United 1, Sheffield United 0.
90:00 +6:43Foul by Jesse Lingard (Manchester United).
90:00 +6:43Martyn Woolford (Sheffield United) wins a free kick in the defensive half.
90:00 +5:51Memphis Depay (Manchester United) wins a free kick in the defensive half.
90:00 +5:51Foul by Chris Basham (Sheffield United).
90:00 +5:39Attempt blocked. Ryan Flynn (Sheffield United) right footed shot from outside the box is blocked. Assisted by Louis Reed.
90:00 +3:56Bastian Schweinsteiger (Manchester United) wins a free kick in the defensive half.
90:00 +3:56Foul by Neill Collins (Sheffield United).
90:00 +2:56 Goal scoredGoal! Goal! Manchester United 1, Sheffield United 0. Wayne Rooney (Manchester United) converts the penalty with a right footed shot to the bottom left corner.
90:00 +1:57Penalty Manchester United. Memphis Depay draws a foul in the penalty area.
90:00 +1:57Penalty conceded by Dean Hammond (Sheffield United) after a foul in the penalty area.
90:00 +0:38Delay over. They are ready to continue.
90:00 +0:37 SubstitutionSubstitution Substitution, Sheffield United. Matt Done replaces Conor Sammon because of an injury.
90:00 +0:15Delay in match Conor Sammon (Sheffield United) because of an injury.
89:05Attempt missed. Memphis Depay (Manchester United) right footed shot from outside the box is close, but misses to the left. Assisted by Daley Blind.
86:22Attempt blocked. Anthony Martial (Manchester United) left footed shot from the centre of the box is blocked.
85:25Anthony Martial (Manchester United) wins a free kick in the attacking half.
85:25Foul by Chris Basham (Sheffield United).
84:49Jesse Lingard (Manchester United) wins a free kick on the right wing.
Foul by Dean Hammond (Sheffield United).
83:15Delay over. They are ready to continue.
81:13Delay in match Bastian Schweinsteiger (Manchester United) because of an injury.
80:47Attempt blocked. Anthony Martial (Manchester United) right footed shot from the centre of the box is blocked. Assisted by Jesse Lingard.
80:18Daley Blind (Manchester United) wins a free kick in the defensive half.
80:18Foul by Conor Sammon (Sheffield United).
79:43Foul by Memphis Depay (Manchester United).
79:43Paul Coutts (Sheffield United) wins a free kick in the defensive half.
77:48 SubstitutionSubstitution Substitution, Sheffield United. Louis Reed replaces Jay McEveley because of an injury.
77:40 SubstitutionSubstitution Substitution, Manchester United. Andreas Pereira replaces Marouane Fellaini.
76:04Delay over. They are ready to continue.
75:19Delay in match Jay McEveley (Sheffield United) because of an injury.
74:41Attempt missed. Bastian Schweinsteiger (Manchester United) left footed shot from outside the box misses to the left. Assisted by Marouane Fellaini.
71:57Corner, Manchester United. Conceded by Martyn Woolford.
69:51Corner, Manchester United. Conceded by Dean Hammond.
68:51Attempt saved. Matteo Darmian (Manchester United) left footed shot from the right side of the box is saved in the bottom right corner.
68:26Anthony Martial (Manchester United) wins a free kick in the attacking half.
68:26Foul by Dean Hammond (Sheffield United).
67:48Attempt missed. Wayne Rooney (Manchester United) header from the centre of the box misses to the left. Assisted by Daley Blind with a cross following a corner.
67:21Corner, Manchester United. Conceded by Paul Coutts.
66:29Attempt saved. Chris Basham (Sheffield United) left footed shot from outside the box is saved in the centre of the goal. Assisted by Martyn Woolford.
65:13Attempt missed. Memphis Depay (Manchester United) right footed shot from outside the box is close, but misses to the left. Assisted by Cameron Borthwick-Jackson.
63:30 SubstitutionSubstitution Substitution, Sheffield United. Ryan Flynn replaces Billy Sharp.
62:52Offside, Sheffield United. Paul Coutts tries a through ball, but John Brayford is caught offside.
59:55Foul by Memphis Depay (Manchester United).
59:55Chris Basham (Sheffield United) wins a free kick in the defensive half.
59:28 SubstitutionSubstitution Substitution, Manchester United. Memphis Depay replaces Juan Mata.
59:15 SubstitutionSubstitution Substitution, Manchester United. Jesse Lingard replaces Ander Herrera.
57:00Corner, Manchester United. Conceded by Jay McEveley.
53:28Foul by Ander Herrera (Manchester United).
53:28Conor Sammon (Sheffield United) wins a free kick in the defensive half.
45:00Second Half begins Manchester United 0, Sheffield United 0.
45:00 +3:08 Half timeHalf Time First Half ends, Manchester United 0, Sheffield United 0.
45:00 +3:01Attempt missed. Daley Blind (Manchester United) left footed shot from outside the box is too high. Assisted by Juan Mata following a set piece situation.
45:00 +2:17Anthony Martial (Manchester United) wins a free kick on the left wing.
45:00 +2:17Foul by Paul Coutts (Sheffield United).
45:00 +0:05Ander Herrera (Manchester United) wins a free kick on the left wing.
45:00 +0:05Foul by David Edgar (Sheffield United).
44:00Foul by Marouane Fellaini (Manchester United).
44:00Martyn Woolford (Sheffield United) wins a free kick in the defensive half.
43:51Attempt blocked. Wayne Rooney (Manchester United) right footed shot from outside the box is blocked.
42:49Hand ball by Chris Basham (Sheffield United).
42:06Delay over. They are ready to continue.
41:05Delay in match Paul Coutts (Sheffield United) because of an injury.
40:34Juan Mata (Manchester United) wins a free kick in the attacking half.
40:34Foul by Dean Hammond (Sheffield United).
36:57Offside, Sheffield United. Neill Collins tries a through ball, but Billy Sharp is caught offside.
30:54Offside, Manchester United. Bastian Schweinsteiger tries a through ball, but Cameron Borthwick-Jackson is caught offside.
26:13Daley Blind (Manchester United) wins a free kick in the defensive half.
26:13Foul by Billy Sharp (Sheffield United).
21:17Corner, Sheffield United. Conceded by Matteo Darmian.
18:35Attempt missed. Wayne Rooney (Manchester United) right footed shot from outside the box is high and wide to the left. Assisted by Ander Herrera.
17:55Hand ball by Conor Sammon (Sheffield United).
11:51Corner, Manchester United. Conceded by John Brayford.
0:19Foul by Marouane Fellaini (Manchester United).
0:19Chris Basham (Sheffield United) wins a free kick in the defensive half.
0:00First Half begins.
0:00Lineups are announced and players are warming up.
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