Villa looked on course for a third successive win after leading at the interval through Christian Benteke's 18th goal of the campaign.
But the Reds equalised through Jordan Henderson early in the second period and then skipper Steven Gerrard's penalty sealed victory for the visitors.
"Our character was outstanding today [Sunday]," Rodgers told Sky Sports 1. "We created chances in the first half, in the second half we upped the tempo.
"It was a real tough game for us.
"Our quality was there as well, and the rhythm and flow of our passing was good."
The quality of Liverpool's equaliser particularly delighted the Anfield boss.
Philippe Coutinho was the architect, his defence-splitting pass from the left channel seized on by Henderson who confidently lofted the ball past Brad Guzan.
"It was a wonderful pass, and Jordan's had a touch and his finish was fantastic because he's got a wee bit of time to think about it," he said.
"I'm really pleased for him and young Philippe has got great, great quality."
Liverpool captain Gerrard felt the visitors did not deserve to be trailing at the break and felt they were rewarded in the second half.
"We felt hard done-by to be trailing at half-time. We thought we should have been at least level," said Gerrard. "We needed to give an extra 10 per cent.
"We had to take a few more risks in their final third and it came off today."
With Liverpool now seventh, three points behind rivals Everton, Gerrard says all the Reds can do now is go out at get three points every week.
["We have to] win every game, finish as high as we can," he said.
"If we can nick fifth or six, that's what we'll try and do."
Villa, who slipped into the relegation zone for the fifth time this season, are level on points with Wigan but have an inferior goal difference.
Manager Paul Lambert said: "I the first half I thought we were excellent, I really did. We were right in.
"In the second half there wasn't much in it. But you can't give goals away like we did - it was definitely a penalty."
"The goal two minutes after the restart was disappointing to lose. But we are playing well enough to win games.
"We are four points from 11th place. There's going to be some twists and turns.
"The thing for myself is that in spells we are playing really well and confidence is a big part of the game of football.
"If you keep that, then that will certainly help you."
Villa have gone 20 games without a clean sheet since the goalless draw with Stoke on December 8.
Lambert said: "If you can get a clean sheet and win 1-0 you would take it. To do that, it will be great."
Asked if Villa can beat the drop, the Scot replied: "Yes, definitely. I think anybody who was here today would say the same. We're certainly playing well enough. We don't look like a team down the bottom at the minute.
"If we going the way we're going, we'll win more games than not."