Javier Hernandez and Wayne Rooney put United ahead 2-0 after just 11 minutes, but substitute Eden Hazard began Chelsea's comeback in the 59th and Ramires equalized seven minutes later.
United manager Alex Ferguson blamed the collapse on fatigue in the wake of Tuesday's Champions League loss at home to Real Madrid. Chelsea's players, though, found the energy only three days after playing in the Europa League in Romania.
"We're lucky to still be in the cup," Ferguson said. "We just ran out of legs and Chelsea were far the better team in the second half. But I can't be critical of my side because ... the emotions and the intensity of Tuesday night took their toll on some of the players."
The semifinals will take place on April 13 and 14, with Wigan now waiting to see whether it will play Blackburn or Millwall, who drew 0-0 on Sunday and have to meet again.
In the Premier League on Sunday, third-place Tottenham missed the chance to extend its two-point lead over Chelsea after losing 3-2 at Liverpool. Newcastle moved nine points clear of the drop zone after beating Stoke 2-1.
United is running away with the Premier League title, holding a 12-point lead over neighbour City. But it was coming off a disappointing loss in the Champions League to Real Madrid on Tuesday and was looking for a quick recovery to stay on course for a cup double.
And Ferguson's team seemed set to give a perfect response when it went ahead after just five minutes against Chelsea, with Hernandez heading past stranded goalkeeper Petr Cech after Michael Carrick pinged the ball over defender Gary Cahill.
Rooney was on target six minutes later, floating a free kick over Chelsea defender David Luiz and United's Jonny Evans, leaving Cech little time to react.
Rooney's goal came at the end of a week in which Ferguson assured the player of his United future despite being dropped from the starting lineup for the Madrid game.
After losing 1-0 at Steaua Bucharest on Thursday in the first leg of the Europa League last-16 phase, Chelsea had looked to be heading out of the FA Cup.
The visiting fans at Old Trafford again turned on interim manager Rafa Benitez and vented their anger over a double substitution early in the second half.
As Frank Lampard and Victor Moses came off, John Obi Mikel and Hazard came on — and the replacements made the desired impact.
Chelsea pulled one back when Hazard curled a shot from the edge of the penalty past goalkeeper David De Gea.
Fernando Torres then replaced Demba Ba, and the substitute striker sent the ball through to Oscar, who fed Ramires for the equalizer.
"I detected some tiredness because the players kept giving the ball away," Ferguson said.
United relied on De Gea's fine save late on to deny Juan Mata a winner.
"It's a day to be proud of the team," Mata said.
United defender Rio Ferdinand could face English Football Association disciplinary action after an altercation with Torres. He shoved Torres after the pair tangled going for the ball late in the match.
There was a pre-match flashpoint, too, with Benitez accusing Ferguson of snubbing his attempt to shake hands.
"I was ready and waiting," said Benitez, who became embroiled in disputes with Ferguson before leaving Liverpool in 2010. "I have some education because I know a lot of people are watching so I know what I have to do."
As Chelsea was playing in the cup, its fight to finish in the top four of the Premier League was helped by Liverpool ending Tottenham's 12-match unbeaten run.
Luis Suarez put Liverpool ahead in the 21st at Anfield with his 50th goal for the club, but Jan Vertonghen headed Tottenham level on the stroke of halftime by meeting a cross from Gareth Bale.
The Wales midfielder set up Vertonghen again after the break as Tottenham went in front, but a defensive mix-up allowed Liverpool to equalize in the 66th as Stewart Downing grabbed a rare goal.
Tottenham defender Benoit Assou-Ekotto then fouled Luis Suarez in the area, and Liverpool captain Steven Gerrard buried the winner from the penalty spot to clinch a third successive win that took his team sixth.
At St. James' Park, Papiss Cisse scored in stoppage time after Yohan Cabaye cancelled out Jonathan Walters' penalty for Stoke to give Newcastle the win that eases any relegation fears.