Captain Steven Gerrard's double booked the seven-time winners' place in the fourth round – where they will face Bolton Wanderers – but they had to withstand some pressure from their hosts.
Fourth-tier Wimbledon levelled through Abebayo Akinfenwa before half-time, but they were guilty of wasting chances while in the ascendancy.
Rodgers admitted his side had taken their foot off the pedal at times, but he felt their second-half performance warranted a place in the next round.
"It was a very tough game, congratulations to AFC Wimbledon - they made it very difficult," he told BBC Sport.
"We started well like we'd want to, controlling the game, but then we lost our shape within the game and we presented them chances by giving the ball away.
"We adapted at half-time but we still had to defend all the way through to the end. The physicality always puts you under pressure - they have good delivery.
"The ball came into the box consistently and of course that can be a threat. You'll never be overly comfortable in those situations.
"I've got to give a lot of testament to my players - they showed a lot of character. In the second half, we should have had two or three more goals."
Gerrard – who announced his intention to leave Liverpool at the end of the season on Friday – was delighted to have scored twice, and he has got his sights set on lifting the trophy for a third time.
"I always enjoy the FA Cup. I've grown up loving the competition," he told BBC Sport. "It is going to be my last time so I want to go all the way.
"Bolton in the next round will be a difficult test. Winning this competition is what dreams are made of and hopefully that will be the case."
