Rudy Gestede and Craig Conway scored late on to seal the win for the Championship outfit, after Chris Taylor had cancelled out Gylfi Sigurdsson's stunning opener.
Swansea had seen Kyle Bartley sent off after a coming together with Josh King early on, before a late challenge on Taylor during second-half stoppage time earned Sigurdsson a red card of his own.
"The lads have stuck together and I thought that really shone through today," Bowyer told Radio Rovers. "There was the early sending off, but after that we were disappointed with the manner of their goal.
"It was a great strike, but we should have closed the ball down better.
"But our response was terrific, Taylor has done fantastically well, it was a great move and a great goal.
"We had to be patient, but fair play to the lads they learnt that today, they stuck at it and were terrific.
"Our organisation, our shape, and the way we pressed at times was great. Then we made the short corner count thanks to Rudy, and Craig's third goal just gave us breathing space.
"We are delighted for everybody, hopefully now we can get another Premier League team in the draw and enjoy a cup run."
