In the post-red shake-up Conte had taken off Boga and brought on Andreas Christensen to keep the base of the team the same shape but there was a striking feebleness to the right side of Chelsea's defence as Burnley went 2-0 up five minutes before half-time.
For Burnley Jack Cork made his debut in central midfield as Sean Dyche packed the centre of the pitch.
The champions had two men sent off in the game, Gary Cahill in the 14th minute and Cescc Fabregas in the 81st minute.
Conte will have some thinking to do, along with the board, as both Cahill and Fabregas will be missing for next week's match with Tottenham Hotspur.
Chelsea coach Antonio Conte berated his players for losing their composure after new captain Gary Cahill saw red on Saturday, and said he would be working on new formations using only 10 men. "As for the transfer market, the club is trying to do the best".
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The champions won 19 of 21 matches across all competitions at Stamford Bridge last season while Burnley took a scant seven points at Turf Moor all season.
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 worldwide. His roaming run into the opposition half, perhaps an attempt to grab the game by the scruff of the neck, was fine, but his lunging tackle into Defour was clumsy and risky and left the referee with no option but to send him off. It really could not have come at a worse time.
"And your first away win is important after last season [not getting one until April], so it kills a few things off early doors". The biggest threat for Burnley is that a bigger club comes in and poaches him.
Batshuayi, who scored the goal which clinched the league title last season, started ahead of new £70million signing Alvaro Morata. They still had the quality to peg Burnley back, particularly as the visitors exhausted as the second half wore on, but lacked subtlety in their approach.
Conte will be hoping he puts in plenty of similar performances this season. The Spaniard then provided an assist for David Luiz to reduce the deficit.