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.
That just left Mendy in between him and an important goal, and after a little body feint, he passed the ball home in front of the delirious United fans.
After almost 50 minutes of being comfortably the second-best team, United looked invigorated, looking a serious threat on the break every time its attackers got the ball.
With just over 20 minutes to go, the game took another twist as Chelsea was awarded a penalty after Aaron Wan-Bissaka kicked the heel of Thiago Silva.
And Jorginho redeemed himself from the spot, calmly slotting home to equalize.
The goal seemed to suck the attacking spirit out of the United team, even after the introduction of Ronaldo off the bench.
Chelsea turned up the pressure, producing chance after chance, but they were unable to find a route through.
Against the run of play, United was handed a glorious opportunity to win the game with just two minutes left, after Mendy's poor clearance fell straight to Fred, but the Brazilian's chip couldn't clear the grateful Mendy.
With the final kick of the game, Chelsea's Antonio Rudiger had a chance to be the hero, but he blazed his volley over the bar.
Chelsea manager Thomas Tuchel called it a "big chance."
"If he maybe hits the target it could have been a fantastic end to a fantastic performance but we have to swallow the result. We played very well and I am happy," he told BBC Sport.
"We were very disciplined and sharp. I am happy with what I saw, it was a pleasure to coach on the sidelines. We can not guarantee results but if we play like this we can increase the chances for results."
Tuchel added: "I am absolutely happy with the performance. We dominated from the first minute, we were aggressive and never stopped attacking. Unfortunately we scored two goals, one for each side."