The hosts had won all four of their previous homes games without conceding but fell behind early on when Pedro finished off a sweeping attack.
The visitors dominated possession and it seemed only a matter of time before they added to their advantage.
However, Burnley’s record signing Robbie Brady – making his full debut following a January move from Norwich – equalised with a stunning free-kick.
The Clarets almost extended their lead before the break but Thibaut Courtois superbly denied Matt Lowton from close range.
Chelsea did not have a single shot on target in the second half as Burnley took the game to the Premier League leaders, Andre Gray testing Courtois with a low drive when he was through one-on-one.
In the end, manager Antonio Conte will perhaps consider this a point gained as the Blues moved 10 points ahead of Tottenham.
Burnley, meanwhile, remain 13th.