Jack Wilshere struck a superb late winner to fire Arsenal past Michael Laudrup's side at the Emirates Stadium.
The England midfielder - out for more than a year with ankle and knee injuries before returning to action this season - capped a man-of-the-match display to finally beat inspired Swansea goalkeeper Michel Vorm with a 20-yard shot on 86 minutes to set up a trip to Brighton and keep alive hopes of Arsenal's first trophy since 2005.
"Jack was outstanding tonight," Arsene Wenger, the Arsenal coach, said afterwards.
"He played in a different position, a bit higher up and it suited him well. I didn't expect him to be this good at this stage.
"I always thought that if we could get Jack back to a reasonable level by January we will have done well, but he is ahead of what I could imagine.
"He is back to what he was before the injury, I think so, yes.
Wayne Rooney, meanwhile, notched an early winner - and missed a penalty - as United saw off West Ham at Old Trafford to earn a fourth-round tie against Fulham.
Rooney paid his respects to his late sister-in-law after netting the game's only goal in the ninth minute.
The England striker also blazed a second-half penalty over the bar - the fifth time United have failed from the spot this season - as the hosts survived a spirited second-half rally from the Hammers.