Despite taking the lead through Steven Fletcher in the 21st minute, the Clarets found themselves behind at the break after Jonathan Macken and Anderson Da Silva both netted.
Owen Coyle's men managed to get themselves back on level terms seven minutes after the break via a Chris Eagles tap-in following some brilliant work from Fernando Guerrero down the flank but it was to no avail as Barnsley's Hugo Colace headed home with 15 minutes remaining.
The loss to Burnley disappointed Coyle, who led his side to the semi-finals of the competition last season.
"We came here to win," he said. "And I know we made six changes to the team from Saturday and including the substitutes we made nine changes. But the team was strong enough to win the game and I think we offered enough to do so.
"You have got to pay credit to Barnsley, any neutral watching it will have seen some good football, end to end, but I'm disappointed to have lost."