The Championship outfit crashed out of the competition on penalties following a 3-3 draw at Brunton Park.
"I'm disappointed because we came here to try and get through to the next round," Sky Sports News quotes Bowyer as saying. "There were two poor pieces of defending.
"We should have been professional and seen it out, and we've got to learn quicker than that. However, full credit to Carlisle. We are extremely disappointed but we have just got to dust ourselves down and focus on the next game."
Carlisle of League One learn who their next League Cup opponents will be tomorrow when the draw for the second round takes place.