The Premier League side must travel to Ewood Park for a sixth round replay after Bowyer's men earned a deserved draw at Anfield with a disciplined and composed performance.
Rovers defended diligently to ensure goalkeeper Simon Eastwood had a relatively quiet afternoon, while Rudy Gestede ensured the visitors carried a robust threat in attack.
"To come here and keep a clean sheet is an unbelievable effort from our players," Bowyer told BT Sport.
"They carried out the plan to a T. We grew in confidence with the chances that we created.
"The big man was a threat for us, Rudy, and I thought he played ever so well.
"Every ball in the box he worried the Liverpool defence. Unfortunately for us, we haven't been able to convert one [chance]."
Bowyer is approaching two years in charge at Blackburn, where he took the reins amid a succession of managerial departures and off-field unrest.
He felt Sunday's showing at Anfield demonstrated the effectiveness of the rebuilding process he has overseen in east Lancashire.
"We wanted to bring people into the club who would fight and show character and try and rebuild the club step-by-step," he said.
"I think you've seen the character of our players. They were terrific. They've given everything for us.
"We knew we'd have to do a lot of defending today. Everyone - they were magnificent today. The prize is they've got to do it again now."
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