Villa striker Gabriel Agbonlahor scored the only goal at Anfield in the ninth minute, hooking home from close range after Philippe Senderos’s header was blocked by Javier Manquillo.
After debutant Adam Lallana and Mario Balotelli shot wide, Liverpool manager Brendan Rodgers sent Raheem Sterling on, but the closest the hosts came to equalising was a shot from Philippe Coutinho that hit the post.
While Villa — who have not lost at Anfield since December 2010 — remain unbeaten this season, Liverpool have now lost twice, following last month’s 3-1 loss at Manchester City, and trail Chelsea by six points.
Villa climbed to second, two points behind Chelsea, who came from behind to beat Swansea at Stamford Bridge thanks to a hat-trick by Diego Costa.
John Terry’s own goal from Neil Taylor’s left-wing cross gave previously unbeaten Swansea an 11th-minute lead, but Costa equalised on the stroke of half-time with a header from a right-wing corner.
He struck twice in 11 minutes to put Chelsea in control, polishing off a Cesc Fabregas cut-back and a mishit Ramires shot, and Loic Remy made it 4-1 with a debut goal before Jonjo Shelvey replied for Swansea.
Brazil-born Spain international Costa now has seven goals to his name and is the first player to score in his first four Premier League games since Coventry City’s Micky Quinn in 1992.
“Diego Costa is one of the best strikers we have at this moment in football,” said Chelsea manager Jose Mourinho.
“The team is an attacking team that creates a lot and creates spaces for a striker. It could have been five or six.”
– Leicester claim first win –
Elsewhere, Everton claimed their first win of the season with a 2-0 success at West Bromwich Albion.
Romelu Lukaku opened the scoring in the second minute against his former club, pouncing on a loose clearance from Jonas Olsson and curling home, with Kevin Mirallas sealing victory in the 67th minute.
An own goal by Harry Kane saw Tottenham Hotspur held to a 2-2 draw at Sunderland.
After Nacer Chadli and Adam Johnson had exchanged quick-fire goals, Chadli teed up Christian Eriksen to restore Spurs’ lead, only for Vlad Chiriches’s clearance to hit Kane and bounce in for an own goal in the 82nd minute.
Graziano Pelle scored twice as Southampton sent Newcastle United to the foot of the table with a 4-0 win at St Mary’s. Jack Cork and Morgan Schneiderlin were also on target.
Leicester City secured their first victory since gaining promotion, winning 1-0 at Stoke thanks to a second-half effort from new signing Leonardo Ulloa.
But fellow promoted side Burnley failed to follow suit, with Scott Arfield seeing a late penalty saved by Julian Speroni in a 0-0 draw at Crystal Palace.
Manchester United host Queens Park Rangers on Sunday, when Radamel Falcao, Luke Shaw, Marcos Rojo and Daley Blind are all in line to make their first appearances for Louis van Gaal’s side.