Late goals from Raheem Sterling and Philippe Coutinho sent Liverpool through after Eidur Gudjohnsen's earlier opener for Bolton, who had Neil Danns sent off when they led 1-0.
And Rodgers was thrilled with his side's late comeback at the Macron Stadium - praising their belief and determination to progress in the competition.
"That was the key as it was always going to be a tough game, Neil Lennon's done a brilliant job here," said the Liverpool manager.
"I thought the players showed brilliant character and spirit.
"We just had to keep our patience - especially against 10 men - you need to keep the width in the game, and we did that.
"Our momentum is still going. Over the last few weeks the character and intensity in the team has all gone up and this is a competition we feel we can do really well in."
Liverpool went closest to breaking the deadlock in the first half when Sterling rattled the bar with a superb curling effort, but it was Bolton who moved ahead when Gudjohnsen converted a penalty in the 59th minute.
Danns' subsequent dismissal for a second bookable offence made Bolton's task tougher, with Jordan Henderson and Emre Can striking the woodwork as the visitors looked to hit back.
Liverpool finally drew level when Sterling volleyed in an exquisite pass from Can after 86 minutes, and Coutinho then fired home from 25 yards in stoppage time to set up a fifth-round tie with Crystal Palace.
"Philippe has the magic to do that and he's a great player. It's brilliant individual skill and a great finish," said Sterling.
"When we hit the woodwork so many times I was thinking it wasn't our day but the boys stuck in really well and got the goals we deserved to win the game.
"The most important thing was to win today's game – hopefully now we'll go to Palace and do the same."
