After a goalless first-half, Jadon Sancho gave the visitors a shock opener on the break, before Jorginho equalised with 20 minutes to go after Thiago Silva was fouled.
Chelsea looked the more likely to find a winner, but they were kept at bay. The result means they are only one point clear at the top of the table.
There was a hint at what was to come after just four minutes. Callum Hudson-Odoi was clean through after terrible defending from Victor Lindelof, but David de Gea produced a great save with his feet.
Chelsea dominated possession and territory, but were limited in how many clear chances they created.
United looked increasingly dangerous on the counter, but Aaron Wan-Bissaka gifted Chelsea a way back into the match.
The full-back kicked through Thiago Silva in the box as the ball dropped from a corner, and the referee pointed to the spot. Jorginho sent De Gea the wrong way to level things up.
Chelsea pushed for the winner and had several shots blocked, before the big chance came with the final kick.