It was a close, yet entertaining match for the first game of the campaign as goals from Kenwyne Jones and Tom Cairney saw two closely matched teams share the points.
Rovers had the better of the possession for long spells but couldn't convert when the chances came their way.
"I thought we were the more dominant side," Bowyer told Radio Rovers post-match.
"I thought we started well and I felt we should have had a penalty in the first minute.
"I was disappointed with the manner in which we conceded the first goal but I thought we recovered well with the standard of football that we played."
After Jones' opener Rovers hit back well and deservedly got back on level terms when Tom Cairney fired home from 20-yards.
"We scored a fantastic goal, it was a great strike," added the Rovers boss. "But full credit to Jordan Rhodes in the build up. He doesn't often get the credit for his hard work.
"We know he puts shift in, as did Rudy tonight, I thought the front two worked ever so hard.
"Credit to Cardiff I thought their two centre halves defended magnificently well. We huffed and puffed and lacked that guile.
"We just needed to be that little bit cuter."