The Premier League leaders had broken the away side's resolve in the 33rd minute when Kai Havertz got on the end of a Reece James cross.
The Blues had ample opportunities to score either side of the German forward's finish, but were unable to find a way past Nick Pope.
And they became the victims of Burnley's smash and grab in the 79th minute as Vydra converted from close range.
The Czech Republic international coolly lifted the ball over Edouard Mendy after fellow substitute Jay Rodriguez had guided Ashley Westwood's whipped cross into his path.
It was the sixth point that Sean Dyche's side had collected from their last seven top flight trips to SW6.
Here are the ratings.
And they became the victims of Burnley's smash and grab in the 79th minute as Vydra converted from close range.
5. Ben Mee 6.5
Puffed out his chest and defended resolutely to keep Chelsea out in the second half. However, over-committed and left the Clarets exposed on occasions after the interval and lax positioning almost cost the visitors more than once. Photo: Ryan Pierse
6. Johann Berg Gudmundsson 6
Couldn't make much of an impact going forward as Rudiger and Chilwell combined to shut down that side of the pitch. Did plenty of running in the opposite direction to help limit the threat of the latter before making way for Vydra in the 70th minute. Photo: Tolga Akmen
7. Dwight McNeil 7
James's advanced role in Chelsea's 3-4-3 meant the winger always had to be on his guard. While his preference would have been to try and isolate Christensen with the ball at his feet at the other end of the pitch, McNeil tracked runs well while working hard on the other side of his game. Photo: Ryan Pierse
8. Ashley Westwood 6.5
A very 'streetwise' display from the Burnley midfielder. Used every trick in the book to slow Chelsea's counters, which eventually resulted in a booking. Knew when to press and when to drop off and whipped in a sumptuous cross for the equaliser. Photo: Alex Pantling