In a absorbing cup-tie at Ewood Park, Rovers were full value as Scott Dann's second half strike cancelled out Alvaro Negredo's goal at the end of the first 45.
Rovers pressed for a winner with Cairney, Campbell and Rhodes all going close in the latter stages, but the draw means their name will be in the hat in Sunday's fourth round draw.
"It might be a bit greedy but we could have nicked it at the end," Bowyer told Radio Rovers.
"I have nothing but praise and pride for this group, it was a terrific performance against a world class team.
"The players are a joy to the football club at the moment. They've really restored some pride, the supporters can go away happy with that.
"We are pleased with how we went about it and how the players responded to going behind. The ages of some of them, it is terrific for them to play in that environment, hopefully it is a motivation for them.
"The fans pay their money and what they want to see is people having a go and working hard and they certainly saw that today from them."
Rovers are back in action next Saturday when Doncaster Rovers are the visitors to Ewood Park. Tickets are on open sale priced from £15 Adults, £10 Seniors and £5 Juniors. Book online today at www.onerovers.co.uk