Christian Benteke put Villa ahead with a crisp strike in the first half but Jordan Henderson equalised early in the second period before Steven Gerrard claimed Liverpool’s winner from the penalty spot.
Defeat left Villa third from bottom in the English top flight, but although they only trail fourth-bottom Wigan Athletic on goal difference, Roberto Martinez’s side have a game in hand.
Villa had been bidding for a third successive win, but instead their momentum deserted them ahead of a run of games that includes daunting trips to Stoke City and leaders Manchester United.
Liverpool, meanwhile, closed to within five points of the European places, although the three teams immediately above them — Chelsea, Arsenal and Everton — all have a game in hand.
“Our character was outstanding today,” Liverpool manager Brendan Rodgers said after the match.
J. Jayalalithaa passes away: Her life in pictures
Photos: Shah Rukh Khan, Deepika Padukone, Alia Bhatt at an awards show
Photos: Big B, Iulia Vantur at Himesh Reshammiya's album launch
Photos: Aamir Khan with 'Dangal' actresses at fashion show
Spotted: Twinkle Khanna, Shriya Saran at an event in Mumbai