Carlisle caused a shock on their first trip to Oakwell since 1986 as they beat Barnsley 3-0 in the first round of the Carabao Cup.
Those alterations looked to have backfired when Carlisle were gifted the opener following a mistake from debutant Aapo Halme.
The defender, signed from Leeds in the summer, let a pass run through his legs for the goalkeeper – only for McKirdy to steal in and prod the ball past a helpless Brad Collins from close range for the 24th-minute opener.
Carlisle set out to inflict more misery on the hosts after the break and doubled their lead when Bridge converted from the penalty spot in the 58th minute after Thomas had been fouled inside the penalty area.
Sheffield United loanee Thomas then added his name to the scoresheet with an arrowed finish in the 64th minute, completing the scoring after bursting into the penalty area.