Incredibly, Vydra made the table-toppers pay for a series of squandered opportunities, coming off the bench and dispatching Burnley's only noteworthy chance of the match to snatch a point for the East Lancashire outfit.
Story of the match
Callum Hudson-Odoi forced Nick Pope into a low save with just a few minutes on the clock and, from a similar position, James worked himself an angle and saw his thunderous effort blocked by Charlie Taylor.
Antonio Rudiger soon went on to miss two headed opportunities and Andreas Christensen also glanced a Hudson-Odoi cross narrowly wide. The Burnley goal was living a charmed life and spectators had only just put their bums on seats!
After just 15 minutes, Chelsea had already racked up five corners and several decent opportunities in and around the box ― a sign of the challenge that Burnley were up against.
The Clarets were struggling to keep up with a mix of penetrative passing through the lines and direct balls up to the wing-backs but committed defending, good goalkeeping, and a little bit of luck was keeping them in the tie.
Chelsea's pressure was relentless and James, in particular, was causing all sorts of chaos from his advanced role on the right.
A low-driven cross was inadvertently deflected goal-bound by Taylor and the Burnley left-back was thankful that Pope's dangling leg managed to keep the ball out.
For all of Chelsea's solid build-up play, it was a relatively simple goal that put them into the lead.
James found himself with plenty of time to pick out a cross and an unmarked Havertz managed to divert a header across Pope and into the corner.
The touchline theatrics provided an interesting sub-plot and it's fair to say that there was no love lost between the two management teams.
Dyche called Tuchel out for the way he confronted the fourth-official, following a decisional dispute with referee Andre Marriner, and that didn't go down too well with the Chelsea representatives.
Later, Steve Stone, a member of the Burnley coaching staff, lost his temper with the antics in the home dugout. "Show some f*cking respect," he shouted to one member of the Chelsea coaching staff.
The hosts slowed down a little after the goal but they came out of the traps firing in the second-half.
Their tempo was much quicker and Thiago Silva almost found reward for that, clipping the outside of the post with a header from close range.
Havertz should have doubled his account but the goal-scorer ballooned his shot over the bar from six-yards after an excellent delivery from Hudson-Odoi.
Hudson-Odoi later found an angle for himself, dancing between three Burnley players, but his drilled effort was stopped by Pope.
There was a growing feeling of frustration around the Bridge but nobody really expected Burnley to capitalise; the visitors weren't in the game. However, against all the odds, complacency did come to bite the Blues on the backside.
Vydra, brought on for Johann Berg Gudmundsson, managed to ghost in at the back post and prod the ball into the back of the net after a header across from fellow substitute Jay Rodriguez. The Chelsea players just stood and watched it happen, almost waiting for VAR intervention...but it never came.
Inevitably, it will go down as a Sean Dyche tactical masterclass: hanging in until the end and letting the substitutes do the business.
David didn't beat Goliath but he left him rather annoyed at full-time!