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 Sir 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, made no comment on the matter but admitted his side were fortunate not to lose.
"We're lucky to still be in the cup," he told MUTV. "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 running they had to do on Tuesday night, against a really top side in Real Madrid, had its bearing on today's result, no doubt about that."
The day's other quarter-final will also be replayed after Championship Millwall and Blackburn Rovers played out a 0-0 draw. Manchester City will play United or Chelsea in the last four and Wigan Athletic meet Millwall or Blackburn.
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
