The win sees Jose Mourinho's men remain in third, but only two points behind leaders Arsenal, while the Red Devils remain seventh, six points adrift of fourth placed Liverpool and a staggering 14 points behind top spot.
Ashley Young nearly put United ahead in the second minute, the winger linked up well with Danny Welbeck before drilling a low effort, which was palmed away by Peter Cech.
Samuel Eto'o gave Chelsea the lead in the 16th minute with a deflected shot from just outside the area, after the striker did well to get past Phil Jones.
Patrice came close to leveling matters in the 27th minute, after the Frenchman dispossessed Branislav Ivanovic, before firing an effort just wide.
Welbeck had a chance to equalise for the visitors in the 37th minute, but the striker's effort from close range was parried away by Cech.
Oscar nearly doubled the Blues lead in the 42nd minute but the midfielder's bicycle kick flew over the bar.
Eto'o doubled Chelsea's advantage in the 45th minute with a finish from close range, following a cross from Gary Cahill.
Eto'o completed his hat-trick in the 48th minute, after Cahill's header was saved by David De Gea, but the Cameroonian was on hand to finish the rebound.
Javier Hernandez pulled one back for the away side in the 78th minute after turning in Phil Jones' shot.
Hernandez came close in the 89th minute but the Mexican's header was directed straight at Cech.
David Moyes' side were reduced to 10 men in the 91st minute after Captain Nemanja Vidic was shown a straight red for a foul on Eden Hazard.