Chelsea had to settle for a point at Stamford Bridge.
Kai Havertz put them ahead with a first-half header from Reece James’ right-wing cross.
The Blues dominated, going close to scoring a second when Thiago Silva’s header from Jorginho’s ball into the box hit the post.
Havertz missed a great chance when he shot over, while Callum Hudson-Odoi was denied by Burnley keeper Nick Pope.
They proved to be costly misses – the Clarets equalised against the run of play with 11 minutes remaining.
It was a well worked goal, with the unmarked Jay Rodriguez nodding Ashley Westwood’s cross back for Matej Vydra to score into an unguarded net.
Chelsea: Mendy, James, Christensen, Silva, Rudiger, Chilwell, Jorginho, Kante (Mount 85), Barkley (Loftus-Cheek 73), Hudson-Odoi (Pulisic 85), Havertz
Subs not used: Kepa, Chalobah, Niguez, Ziyech, Azpilicueta, Sarr.