Cristiano Ronaldo scored his 17th goal of the season to earn Manchester United a point after David de Gea had frustrated Chelsea with several superb saves.
The visitors were thoroughly dominant in the first half at Old Trafford but were unable to find a way past De Gea, who made saves from Kai Havertz and N'Golo Kante.
But Chelsea finally broke the deadlock midway through the second half when Marcos Alonso volleyed home from an angle.
The Blues' celebrations had barely finished as Manchester United responded within a minute, Nemanja Matic playing a brilliant ball into the box and Ronaldo taking a touch before thumping a shot into the net.
That brought Old Trafford to life - there had been a subdued atmosphere inside the ground up to that point, while discontent against the club's owners was still clear as protests were held outside for the second home game in succession.
Reece James hit the outside of the post while Ronaldo could have secured all three points for the hosts but his effort from a difficult angle was saved by Edouard Mendy.
But the game ended all square as Chelsea edged closer to securing a top-four finish. Thomas Tuchel's side are third, eight points ahead of fifth-placed Tottenham with five games remaining.
Manchester United are sixth, five points behind Arsenal in fourth but having played two games more.
Source: BBC