Chelsea, who had club captain Gary Cahill and Cesc Fabregas sent off, trailed 3-0 at the interval, and despite finding a route back into the match in the second period, Burnley held on for their first ever Premier League victory at Chelsea.
The big team news from the Chelsea camp was that there was a first-team debut for Ivory Coast attacker Jeremie Boga, who spent last season on loan at Granada. The 20-year-old was handed a start in the final third alongside Willian and Michy Batshuayi, with Alvaro Morata only on the bench. Antonio Rudiger, meanwhile, made his Premier League debut.
It was the away team that had the first half-chance of the match inside three minutes, but Ben Mee's header from a Robbie Brady free kick never had enough power to trouble Thibaut Courtois between the Chelsea sticks.
Chelsea's first sight of the Burnley goal arrived in the 11th minute when Willian appeared to find space inside the penalty box following a deflection, but James Tarkowski came from nowhere to make a brilliant clearance.
Chelsea were starting to force the issue after a slow start, but the champions were reduced to 10 men in the 14th minute when captain Cahill was handed a straight red card for a late challenge on Steven Defour.
Burnley actually had the ball in the back of the Chelsea net in the 19th minute when Vokes headed home from close range, but the goal was disallowed, much to the relief of the home side's supporters.
Vokes was not to be denied five minutes later, however, with the Welshman turning a Matthew Lowton cross into the bottom corner as the champions were stunned on home soil.
Chelsea did their best to launch a positive response, but the Blues were struggling, and Burnley incredibly doubled their lead in the 39th minute when Stephen Ward broke into the home side's box before finding the top corner with a stunning strike.
It was to get worse for the champions in the first half, though, as Vokes made it 3-0 to Burnley in the 42nd minute when the striker headed a wonderful Defour cross into the corner of the net.
Chelsea's first effort of the second period came from Rudiger, but the summer arrival from Roma did not trouble Tom Heaton with a long-range strike. The Burnley goalkeeper was called into action moments later, however, as he dived to keep out a Marcos Alonso strike.
Vokes had a decent chance to complete his hat-trick in the 56th minute after again reacting first to a high ball inside the Chelsea box, but the striker's header just dropped wide of the far post.
Morata came off the Chelsea bench just before the hour as the club-record signing replaced Batshuayi, although there was no sign of Diego Costa following the striker's high-profile falling-out with Conte.
Chelsea came close to registering in the 61st minute when Alonso bent a free kick towards the top corner, but Heaton was on hand to make another brilliant save as Burnley kept their three-goal lead intact.
Brady might have made it four for Burnley in the 66th minute after meeting a deep cross from Defour, but the midfielder's effort was wide of the post after Chelsea's defence had gone to sleep once again.
The home side were back in the match in the 69th minute when Morata headed a Willian cross into the bottom corner to register his first goal for the club, although the champions still had a lot to do to turn the game around.
Chelsea thought that they had scored a second in the 73rd minute when Morata turned a low Christensen effort into the bottom corner, but the offside flag saved Burnley as the pressure increased.
N'Golo Kante missed the post with a long-range strike 10 minutes from time as Burnley dropped deeper and deeper, but Chelsea's challenge all but ended in the 81st minute when Fabregas picked up a second yellow card for a poor challenge on Cork, leaving the home side with nine men.
The Blues managed to score a second in the 88th minute when David Luiz converted a flick from Morata, but Burnley, who hit the post late on through Brady, held on to record all three points following an incredible game at Stamford Bridge.
Next up for Burnley is a home game against West Bromwich Albion next Saturday, while Chelsea face a trip to Tottenham Hotspur on Sunday afternoon.