Chris Wood continued his scoring streak to spare the Clarets a trip to Oakwell and an FA Cup third-round replay on an afternoon of penalty drama.
Burnley were denied one spot-kick by the intervention of VAR but then in stoppage time were awarded another which Wood coolly converted to make it three goals in as many games and six for the season.
The striker came off the bench to bag the winner after Burnley boss Sean Dyche had made eight changes to his starting line-up following back-to-back Premier League wins.
There were welcome returns from injury for Nick Pope, Steven Defour and Stephen Ward and valuable game time for others.