Leon Best's brilliant volley with a minute of normal time remaining drew Rovers level after Johnny Russell's first half penalty, but Bowyer was adamant his side's overall display deserved more.
After a confident start, Rovers were disrupted by a serious shoulder injury to Ruben Rochina, and striker Jordan Rhodes had two penalty claims rejected late in the game.
"I thought we deserved three points, to be perfectly honest," Bowyer told Sky Sports.
"I thought we were magnificent today in the way we passed the ball in the first 20 minutes. The injury to Rochina just broke our stride, but looking at how we responded in the second half, I can't fault the effort from the players.
"The spirit and desire that people have questioned in the past came through today. I'm delighted with them, and a little bit sorry for them that they only get one point because they deserve more."
While Rhodes was frustrated at seeing two penalty claims waved away by Stuart Attwell, but Bowyer was rueful that his side did not get the decisions.
"Perhaps the referee can't give too many. We had a couple of great shouts for penalties, but we have to concentrate on how well we played. There were some great debuts, I thought Tommy Spurr and Alan Judge were magnificent."