Despite dominating the opening 45 minutes, Chelsea were unable to break the deadlock and so the two sides were level going into half-time.
And it was actually United who scored the first goal of the game five minutes after half-time.
A disastrous touch by Jorginho set Jadon Sancho clear and the Englishman made no mistake when one-on-one with Edouard Mendy.
Jorginho was able to redeem himself 18 minutes later.
The Blues were given a penalty when Thiago Silva was taken out by Aaron Wan-Bissaka inside the box.
The Italian midfielder stepped up and he made no mistake as he restored parity.
United had a golden opportunity to win it in the 88th minute through Fred.
The Brazilian was gifted a golden opportunity after a mistake from Mendy.
But Fred could not take advantage. He attempted to chip the ball into the net but got his effort horribly wrong and Mendy was able to gather easily.
Oh no, Fred. In truth, he had a pretty good game.
But he may well be annoyed at himself for not giving his side all three points.
Antonio Rudiger also had an opportunity to win it for his side with virtually the last kick of the game
But he failed to get his shot from close range on target and thus the game ended 1-1.
Will Ralf Rangnick be successful at Manchester United? | The Football Terrace
Chelsea remain at the top of the Premier League table but their lead has now been cut to just one point after Man City beat West Ham earlier on Sunday.
United, meanwhile, are eighth and five points off the top four.
Ralf Rangnick is expected to be appointed manager in the next few days. He will be looking to build on the Red Devils’ battling point at Stamford Bridge.