With a Wembley semi-final against Manchester City on April 14 the prize for the victors, runaway Barclays Premier League leaders United made seven changes from the side which won at Sunderland on Saturday, with Robin van Persie on the bench and Wayne Rooney not in the matchday squad.
Chelsea, seeking a fifth FA Cup in seven seasons, also made seven changes from the XI which began the loss at Southampton, with John Terry and Frank Lampard dropping to the bench and Demba Ba starting up front.
The first-half was played largely in third gear, with neither goalkeeper overly tested.
Javier Hernandez's swirling long range shot was the closest for United, which Petr Cech managed to turn away with his legs. At the other end, Ba tested David De Gea with an angled drive, which was beaten away by the Spaniard.
It took Chelsea just three minutes after the break to break the deadlock thanks to a stunning goal from Ba.
Juan Mata collected the ball in midfield and lifted a tidy pass to the onrushing Ba, who met it spectacularly to volley over and past De Gea to give the hosts a 1-0 lead.
On the hour mark, United threw on Van Persie for Tom Cleverley just moments after Cech had produced one of the saves of the season to deny Chicharito's goal-bound header.
Five minutes later Ryan Giggs was on for Nani as United's bench sensed the Cup tie slipping away.
But the substitutions had little effect as apart from a half chance for Van Persie, it was Chelsea which finished the stronger as Mata and Eden Hazard both carved out multiple chances for themselves to keep United on the back foot.