United started well and took the lead on 11 minutes when as Juan Mata marked his 31st birthday with a goal against his former club.
But Chelsea were level by half time after an error from United goalkeeper David de Gea, who spilled a long-range shot from Antonio Rudiger into the path of Marcos Alonso, who gobbled up the rebound.
Both sides lost centre backs to injuries during the second half as Antonio Rudiger and Eric Bailly limped off.
MATCH FACTS AND STATS
Manchester United: De Gea; Young, Bailly (Rojo 71), Lindelof, Shaw; Herrera, Matic, Pogba; Mata (McTominay 81); Lukaku, Rashford (Sanchez 64)
Subs not used: Martial, Pereira, Darmian, Romero
Goal: Mata 11
Booked: Herrera, Rojo, Young
Chelsea: Kepa; Azpilicueta, Rudiger (Christensen 66), Luiz, Alonso; Kante, Jorginho, Kovacic (Loftus-Cheek 75); Willian (Pedro 84), Higuain, Hazard
Subs not used: Zappacosta, Giroud, Barkley, Caballero
Goal: Alonso 43
Booked: Willian, Kovacic