Mohamed Salah's coolly taken penalty for Liverpool ensured Aston Villa boss Steven Gerrard's much-heralded return to Anfield ended in defeat.
Tyrone Mings was penalised midway through the second half for a foul on Salah, who took the spot-kick himself for his 14th Premier League goal of the season.
Liverpool were earlier kept out by keeper Emiliano Martinez, while Salah missed a great chance to add a second.
The win keeps Jurgen Klopp's side a point behind leaders Manchester City, who beat Wolves 1-0 earlier on Saturday.
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