The goal was the first Chelsea have conceded this season from open play.
"Of course I'm disappointed. If we played this game 100 times, we'd win 99 times. Today we didn't win it. The second goal would have decided it. It's always possible you miss one ball, one cross, and this can happen," Chelsea manager Thomas Tuchel said after the game.
"It's our fault. We should have scored. We created so many chances.
"You have the opponent where you want to have them. You create so many chances. We let them believe it was possible to steal a point by pure luck and that's what happened.
"This is not the first game like this and won't be the last. I was impressed by the performance, the attitude, the quality. It's football, it was against us today."
Chelsea had dominated possession throughout the match with Nick Pope continually being call on to make impressive saves.
However, Kai Havertz's first-half strike was the only thing that separated the sides.
Reece James floated a cross into the box on 33 minutes which Havertz headed into the net.
Burnley had few opportunities to attack but only needed one to find their equaliser.
Jay Rodriguez nodded a cross down into Vydra's path and he hit it into the net from the centre of the box.