The hosts, knocked out of the Champions League by Real Madrid on Tuesday after a controversial red card for Nani, took a fifth-minute lead through Javier Hernandez and soon doubled it with a Wayne Rooney free kick.
Just when the Premier League leaders had started to smell a Wembley date and some comfort after a horrible week, Chelsea made a double substitution count and pulled one back on 59 minutes through Eden Hazard before Ramires levelled on 68.
Chelsea had a golden chance to win the game in the dying seconds but United keeper David de Gea brilliantly stuck out a foot to divert Juan Mata's powerful effort, leaving the Londoners disappointed that they had not snatched victory.
"We deserved to win, the first half we made two mistakes, we conceded two goals but after we had some good chances in the counter-attack and in the second half we were on top of them," Chelsea interim manager Rafael Benitez told a news conference.
There is no date yet for the replay as Chelsea have a Europa League last-16 second leg to play against Steaua Bucharest this week but when they do meet again it will be amid fresh tension between Benitez and United counterpart Alex Ferguson.
The Spaniard accused the Scot of failing to shake hands with him before the match.
"I was waiting at the beginning ... I have education because I know a lot of people are watching so I know what you have to do," he said.
Ferguson, who did not give a post-match news conference, did not comment on the matter.
The day's other quarter-final will also be replayed after Championship (second division) Millwall and Blackburn Rovers played out a 0-0 draw. Manchester City and Wigan Athletic booked their places in the semi-finals on Saturday.
SWIFT REWARDS
Still smarting from their cruel Champions League exit, United sought to quickly soothe the pain and started the match fired up for the task at a chilly Old Trafford.
They struck twice within the first 11 minutes, Hernandez opening the scoring when midfielder Michael Carrick's long ball over the top found his head and he nodded the ball neatly over stranded keeper Petr Cech.
The Mexican poacher has now scored in six of his last seven games against Chelsea, who have been the most successful team in the FA Cup in recent years having won it four times in the last six seasons.
Ferguson's men doubled their lead six minutes later after winger Nani was fouled by Victor Moses to win a free kick just outside the area on the left.
Rooney, who faced media speculation over his future when he was dropped to the bench for the Real Madrid game, stepped up to swing the ball over the top of everyone and into the far corner.
Chelsea, who suffered their own European troubles this week with a 1-0 defeat by Steaua Bucharest in the first leg of their last-16 tie, offered little in terms of attack in the first half while United looked dangerous whenever they went forward.
But that all changed in the second half with Benitez, who had earlier heard both sets of fans chanting "You'll be sacked in the morning", making an early double substitution that yielded swift rewards.
Eden Hazard and John Obi Mikel came on for Frank Lampard and Victor Moses, with Benitez saying changes had always been planned because he wanted to start the match with fresh legs after Thursday's European game.
Seven minutes after his introduction, Hazard got on the end of a pass by Mata to unleash a right-footed strike from the edge of the area which flew in to ignite a Chelsea recovery.
Nine minutes later, Demba Ba fed Oscar who played in Ramires with the Brazilian finding the net with an accurate left-footed finish to stun United and even some of Chelsea's fans who had been voicing their discontent earlier in the game.
Chelsea ended the match looking the more likely winners and United had De Gea's foot to thank that they did not end the week with just one piece of silverware still to fight for.
The visitors celebrated as if the draw had been a win and will take confidence from the performance ahead of the replay, although Benitez said it was too early to think about Manchester City, their semi-final opponents if they get through.
