The Reds were staring down the barrel of back-to-back defeats for the first time under Rodgers when Christian Benteke fired relegation-threatened Villa into a 1-0 lead at half-time.
But the response from Rodgers' men was emphatic; Jordan Henderson stylishly levelling two minutes after the break before Steven Gerrard sealed the points from the penalty spot.
An elated Rodgers told Sky Sports after the match: "Our character was outstanding today. I thought we played reasonably well in the first half but we made a mistake and were punished by a good finish from Benteke.
"I thought we created chances in the first half and second half we upped the tempo and going forward we were excellent. Defensively we were nice and solid so it was a terrific win for us. Paul's (Lambert) teams always go right to the end and it was tough for us.
"Our quality was there as was our drive and determination. I thought that tactically we adapted very well and overall I'm very, very pleased. After losing the previous game (against Southampton) you want to bounce back and we did that."
Rodgers reserved special praise for midfield duo Philippe Coutinho and Jordan Henderson after the pair linked up clinically for Liverpool's equaliser.
He added: "The pass (from Coutinho) and the actual run from Henderson, his determination and desire to go and run forward was outstanding. A wonderful pass and fantastic finish. I'm really pleased for him and young Philippe has great, great quality."