The champions were on the back foot after just 14 minutes when they were reduced to 10 men as Gary Cahill was shown a straight red card for a lunge on Steven Defour.
Moments later Vokes had the ball in the Chelsea net from an offside position after a bout of headed ping-pong as Chelsea struggled to rejig.
Any hopes of a recovery were dented when Cesc Fabregas was sent off after collecting a second booking late on, although David Luiz did set up a grandstand finish with a powerful volley to make it 3-2 with two minutes left.
Chelsea will now be without both Cahill and Fabregas for their next Premier League game - a trip to Wembley to take on Tottenham Hotspur next Sunday.
It has been a baptism of fire for the former Real Madrid striker, who missed a penalty in the Community Shield defeat against Arsenal last week, but Conte was full of praise for the Spain global.
"In the next game, I don't know if Pedro is able to recover from his injury", Conte said.
The Blues trailed Burnley by three goals in the opening 45 minutes - and although they mustered a comeback in the second half, they only succeeded in making the scoreline look respectable at 2-3.
The Brazilian curled an exquisite 69th-minute cross into space behind the Burnley defence and substitute Morata stooped to guide a diving header past Heaton, just nine minutes after coming on. A quick free-kick led to Defour swinging in a cross from the right and an unmarked Vokes planted a header past Courtois, with loud boos greeting the champions when the half-time whistle was blown.
Chelsea boss Antonio Conte joked that he will prepare his side to go down to 10 men for the next game.
Chelsea brought in Alvaro Morata in the second half and it paid off as the Spaniad got his first goal for the club to make it 3-1 in the 69th minute of the game.
Chelsea had their first real chance at goal 11 minutes in when Cesar Azpilicueta's low cross was deflected and almost sat up for Willian to fire home in the centre of the goal.
There was still time for more drama as David Luiz's goal made for a nervy finale, but Burnley held out to claim all three points.