After Javier Hernandez opened the scoring in the fifth minute, Wayne Rooney marked his return to the United starting line-up with another early goal to put United 2-0 up at half-time.
Alex Ferguson's side looked poised to move on from the hurt of their Champions League exit to Real Madrid, but second-half goals from Eden Hazard and Ramires meant Chelsea kept one hand on the trophy.
The visitors might even have won, had David de Gea not produced a fine late stop to thwart Juan Mata, and Chelsea interim manager Rafael Benitez said his side would have been worthy victors.
"Especially in the second half we were on top of them, created the best chances and were controlling the game," he said.
"Juan Mata told me that David de Gea made a great save, so it was unlucky but a great save."
United were guilty of an oddly disjointed performance, but Benitez - barracked, once again, by his own fans - could point to a second-half double substitution that changed the game's momentum.
After the pain of Tuesday's loss to Madrid, Ferguson spared himself further unwanted headlines by recalling Rooney to his starting line-up, with Robin van Persie on the bench.
Benitez resorted to what has emerged in recent weeks as his strongest team, meanwhile, with Fernando Torres and captain John Terry both relegated to the bench.
The two clubs' fans had only just started to run through their repertoire of songs about each other when United took the lead.
Michael Carrick sent an angled pass over the Chelsea defence and as visiting goalkeeper Petr Cech rushed out, Hernandez guided a gentle header over him and into the net.
Six minutes later it was 2-0 as Rooney announced his return with a goal.
Victor Moses was penalised for a foul on Nani near the left-hand apex of the penalty area and from the free-kick, Rooney bent in a cross that eluded everyone to creep inside the far post.
Jonny Evans blocked from Demba Ba and Frank Lampard shot straight at De Gea as Chelsea sought a response, but United were soon back on the attack.
After Hernandez had diverted a Nani cross wide of the near post, Cech produced a magnificent double-save.
An instinctive block from Rooney left the Chelsea goalkeeper on the floor but he sprang to touch the ball over the bar after David Luiz inadvertently headed towards his own goal.
Cech saved again, from Tom Cleverley, but United grew sloppy, and also lost Nani to injury just before half-time.
Having been controversially sent off against Madrid, the Portuguese winger had to leave the fray early for the second game in succession after abruptly going down on the edge of the Chelsea box.
Rooney feathered a chip over the bar in the injury time that followed but United's carelessness had given Chelsea hope for the second period.
Benitez made a quick re-shuffle shortly after the interval, introducing Hazard and John Mikel Obi in place of Victor Moses and Lampard.
The move drew chants of 'You don't know what you're doing!' from Chelsea's fans, but it yielded immediate results.
Hazard reduced the arrears with a glorious strike in the 59th minute, ghosting into the box from the left and shaping a gorgeous curler into the right side of the goal.
In response, Ferguson introduced Van Persie, but Chelsea's tails were up and in the 68th minute they drew level.
A counter-attack left United's defence exposed and when Oscar fed the ball wide to Ramires, he rolled the ball inside to throw Evans off balance before stroking a shot into the bottom-left corner.
Chelsea had two opportunities to win the game at the death but de Gea saved brilliantly from Mata in the last minute before parrying a fierce shot from Hazard in added time.
The winners of the replay, the date for which has not been announced, will play Manchester City in the semi-finals.