Chelsea was completely dominant in the first half, striking the bar and forcing United goalkeeper David De Gea into numerous saves.
But after the halftime break, Jadon Sancho pounced on a poor touch from Jorginho to break through and slide past Edouard Mendy to give United a surprise lead.
Chelsea did level midway through the second half as Jorginho redeemed himself, coolly slotting home a penalty.
The draw means Chelsea moves a point clear of Manchester City at the top of the table, while Manchester United move up to eighth.
However, after the game, De Gea said "it is not enough to draw."
"We already improved against Villareal, we won, qualified and now we played a very good team away from home. The effort of the team was very good," he told Sky Sports. "They had the best chances. For me it is not enough, a draw. We are in a tough moment. I am proud of the effort of the team.
"On the pitch, in the goal, I was feeling danger for nearly the whole game. We defended well, they missed big chances and the one they give us a chance we score. It is not enough to draw but at the moment it is a big point.
"The last games before we were conceding three or four goals. We are improving. This is just two games. At least now we are defending well. We are fighting for every ball. We are a team."
To and fro
After Ole Gunnar Solskjaer's recent sacking, Michael Carrick was appointed as the interim manager while the club searched for another interim manager to see it through to the end of the season.
And although reports have suggested Ralf Rangnick is set to be named the new coach, Carrick was in the dugout at Stamford Bridge.
The former Manchester United and England midfielder won his first game in charge, beating Villarreal in the Champions League on Thursday.
And ahead of United's clash against table-topping Chelsea, Carrick made the key decision to drop club talisman Cristiano Ronaldo as he opted for a defensively minded team.
In the first 45 minutes, despite having the majority of the possession, Chelsea was unable to break through.
The reigning Champions League winner was completely dominant but found De Gea in excellent form, producing an excellent save from Callum Hudson-Odoi to deny the hosts.
And five minutes into the second half, the breakthrough was made, but not by Chelsea.
A high clearance was sent into the London sky as United cleared a corner. Jorginho's touch just rolled away from him enough allowing a sprinting Sancho to nick the ball off him.