Goals from Javier Hernandez and Wayne Rooney put United in a commanding position before the break, but substitute Eden Hazard and Ramires replied for the Blues to take the tie to a replay at Stamford Bridge.
The home side went ahead with five minutes on the clock when Hernandez's looping header found the net following Michael Carrick's searching pass.
Their advantage was doubled six minutes later as a Rooney free kick out on the left evaded everyone to fly in at Petr Cech's back post.
Chelsea cut the deficit on 58 minutes through Hazard's curled finish from the edge of the area.
They were level 10 minutes later when Ramires was allowed to surge forward before sweeping a left-footed shot past David de Gea.
Juan Mata had a chance to win it for the visitors in the 90th minute, but his effort was saved at point-blank range by the leg of De Gea.