The Clarets are now without a win in eight games across the board and have won just twice in 23 games in all competitions.
Ian Woan took command at Turf Moor, with boss Sean Dyche absent having contracted Covid, as the Terriers arrived for the third round tie.
Jay Rodriguez opened the scoring in the 28th minute when beating Ryan Schofield to Ashley Westwood's pinpoint pass.
The 32-year-old headed home his fifth goal of the campaign having netted four in Burnley' s triumph over Rochdale in the League Cup back in September.
However, the Clarets failed to build after drawing first blood and it was the visitors who almost pulled themselves level before the break.
Jonathan Russell climbed above Ben Mee to meet Daniel Sinani's set-piece only for the midfielder's header to come back off the post.
But the away side capitalised on a laboured second half display from Burnley to book their place in the fourth round of the competition.
The Championship side equalised in the 74th minute when Matt Lowton was guilty of giving away possession on the halfway line and Josh Koroma tapped in substitute Sorba Thomas's cross from close range.
Town then turned the tables with three minutes remaining when Matty Pearson found space in-between James Tarkowski and Jack Cork to head Sorba's corner into the back of the net.
Burnley could've restored parity late on, but Tarkowski missed the target with a free header from Westwood's free kick.