Rovers' 34-year unbeaten record against their near neighbours looked in jeopardy after Junior Stanislas' 78th minute stunner put the home side in front, but Jordan Rhodes was in the right place at the right time to make the most of Michael Duff's clearance as the ball deflected into the back of the net to equalise six minutes from time.
Speaking after the match Rovers boss was delighted with how his players coped with the white hot atmosphere of an East Lancashire derby.
"We have a young squad and the average age today in the starting eleven was 22, coming into a hostile environment," said Bowyer.
"But they leave with their heads held high because they took everything that Burnley threw at us in the first half and grew into the game.
"I thought our second half performance certainly warranted a point, the goal they scored was a worldy against the run of play, but then our goal was a bit lucky.
"The age of the team, the resilience, the character and the spirit that they showed, we leave here very proud.
"We are building for the future and that is a massive day for them in terms of growing.
"Today you can't single anybody out, the goalkeeper kept us in it early doors and I'm delighted for Jake as he's made a mistake or two but that shows how well he has bounced back.
"They are young lads, they are going to make mistakes and it is how they react to that, I think they've done tremendously well today."