In a match that failed to sparkle in the opening half, it was United who took the lead early in the second half through Jadon Sancho, but a Jorginho penalty leveled matters after Thiago Silva was fouled inside the box by Aaron Wan-Bissaka.
Chelsea could have taken a four-minute lead when Callum Hudson-Odoi missed a clear opportunity when put through on goal, but his shot was expertly saved by David de Gea.
The remaining half was largely uneventful, with Chelsea controlling much of the possession as United sat back deep and waited.
Antonio Rudiger came closest to giving the Blues the lead just after the half-hour mark after latching on to a sweet pass on the edge of the box and unleashed a brilliant shot towards the goal, but he hit the crossbar.
Hudson-Odoi then blew a good opportunity after he created a lot of space around him and fired a terrific shot from the edge of the box towards the bottom left corner, but the goalkeeper made a brilliant save and stopped the ball from hitting the net.
Man United felt they should have received a red card just before the break when Fred went down inside the box, but the referee waved their pleas away.
United then took the lead against the run of play five minutes into the second half when Sancho took advantage of a Jorginho poor touch to race clear and finish past Edouard Mendy.
Timo Werner then wasted a clear chance in the 58th minute, shooting inches of the right post after finding some good space inside the box.
The hosts however got the chance to equalise from the spot when Thiago Silva went down inside the box, and Jorginho easily beat David de Gea.
Despite the Blues piling pressure on Michael Carrick’s team, they could not find the winner as they settled for the point.