Rodgers' men returned to Anfield for the third time in eight days following impressive Premier League wins over Manchester City and Burnley.
But they turned in a lacklustre first-half display as their Championship opponents produced a disciplined performance.
Kolo Toure struck the post during the second period and Rodgers still reserved praise for his players, even though a late rally failed to avoid the need for a replay at Ewood Park.
"I give credit to Blackburn, they've come here and defended well, and I've got to give a huge amount of credit to my own players," he told BT Sport.
"We weren't at our best and we had two tough games leading into it.
"We probably just missed the sharpness in our game
"But we kept going, we kept trying to create chances to get that goal from the beginning right to the very end."
Referee Andre Marriner was unmoved by a succession of marginal Liverpool penalty appeals and Rodgers believes spot-kicks should have been awarded for Matt Kilgallon's 19th-minute challenge on Adam Lallana and when a set-piece struck Blackburn forward Rudy Gestede on the arm.
"I thought we should have had a penalty - Adam Lallana one was a clear penalty," he said.
"The defender is recovering back in, he's on the wrong side and he can't get to the ball.
"You could argue when Gestede handballs it off the header from the free-kick that we could have had another penalty.
"Sometimes you need that little bit of luck."
Gestede was involved in an unfortunate early incident when Liverpool defender Martin Skrtel went up for an aerial challenge with him and landed awkwardly.
Skrtel briefly lost consciousness and required lengthy treatment before leaving the field on a stretcher - at which point Rodgers confirmed the player asked to come back on.
"He was shouting on the stretcher, that was what he wanted to do," he added.
"Obviously you've got to leave that with the specialists and the medical team and he had to come off.
"I think he'll be fine, he'll be able to play in the next game."
