The FA Cup holders Arsenal came from one goal down to beat Sunderland 3-1 in their third round encounter at the Emirates Stadium.
They were joined by both Manchester clubs for the next round. United looked destined for an unwanted replay until Wayne Rooney struck in the 93rd minute after a foul on substitute Memphis Depay.
It was only the second shot on target that United had mustered in the whole game, against a team 47 places below them in the English league system. Louis van Gaal must have breathed a huge sigh of relief but will no doubt face a lot of criticism. Credit must be given to Sheffield United in the way they defended at Old Trafford.
Wayne Rooney took to Twitter to post a photo of his celebration of the winning goal.
Manchester City, meanwhile, thrashed Norwich City 3-0 to progress further.
Earlier, Arsenal manager Arsene Wenger fielded a relatively weaker XI with Nigerian Alex Iwobi getting a start in London. The 19-year-old forward partnered Theo Walcott and Joel Campbell in a three man midfield just behind Olivier Giroud, thus giving Mesut Ozil a much needed rest.
Calum Chambers and Alex Oxlade-Chamberlain also got a start, as also for Kieran Gibbs. Laurent Koscielny, Gabriel and Hector Bellerin complete the outfield line-up with Petr Cech keeping his place under the bar.
After a nervy start to the match, Sam Allardyce’s side took the lead with Dutch forward Jeremain Lens scoring from an opportunistic position to send the packed Emirates Stadium to silence. But the Black Cats found themselves conceding within eight minutes as Costa Rican Campbell scored from a Walcott delivery.
The second half started with the Gunners taking the initiative, but it was the double substitute in the midfield with Aaron Ramsey and Mikel Arteta coming on for Iwobi and Chambers, which changed the course of the match.
Ramsey took just five minutes from the change to score the second goal for his side, from a Ballerin run, in the 72nd minute. Then, the Spaniard workhorse set-up Giroud to score the third goal, second in a space of five minutes.
Elsewhere, fellow Premier League clubs Stoke City, Bournemouth, Everton and Watford register their respective wins, even as Aston Villa continued their poor form with a 1-1 draw with Wycombe earlier.
In an all Premier League club clash, Crystal Palace beat Southampton 2-1 in London.