Aston Villa 0-1 Liverpool
Daniel Sturridge was the match-winner for a second straight week as Liverpool won at Villa Park to make their best start to a season for five years.
The Liverpool striker finished off a flowing first-half move to score his eighth goal in seven league games.
Villa were outplayed early on but improved in the second half, with Liverpool goalkeeper Simon Mignolet twice denying Christian Benteke.
Both sides had late penalty appeals waved away by referee Mark Clattenburg.
Liverpool last won their opening two matches of the season in 2008-9 and this victory means Brendan Rodgers's side extend their unbeaten run to 10 Premier League matches, a streak that began with a win at Villa Park in March.
The platform for victory was set during a first half in which the visitors were utterly dominant. Villa improved markedly after the break but the damage was done.
It has been a brutal start to the season for Paul Lambert's young side. During their magnificent victory at Arsenal and contentious defeat at Chelsea, they displayed tangible signs of progress but against Liverpool they looked flat and, perhaps, fatigued by those two demanding trips to London.
Kolo Toure did well to contain Benteke to just a handful of chances, while Gabriel Agbonlahor only threatened in fits and starts. But Villa's defensive problems remain - this was their 25th successive match without a clean sheet, a record they will look to put right in their next league match against Newcastle in three weeks' time.
Much of the credit must go to Liverpool, though, who once again displayed why there is a growing sense of optimism around Anfield.
The win over Stoke had been Liverpool's first on the opening day since they last mounted a credible title challenge five years ago. It was in that season they last won their opening two matches of a campaign, something they had only done twice in the previous 18 seasons.
There were more signs of encouragement at Villa Park.
Liverpool bossed the opening 45 minutes with a swagger and fluidity that, at times, Villa were unable to contain. Midfielders Steven Gerrard and Lucas Leiva were imperious and Philippe Coutinho was a constant threat as Liverpool's frontline swapped places at will.
His understanding with Sturridge borders on the telepathic. That had been obvious against Stoke and it was that link-up which created Liverpool's opening goal from a move that began with Jose Enrique wide on the left. Coutinho allowed the Spaniard's low cross to run across him and into the path of Sturridge. The former Chelsea striker took the ball around Antonio Luna, beyond the sprawling Brad Guzan and poked it high into the roof of the net with his left foot.
Starved of possession and with the home crowd beginning to grow restless, Villa might have wilted. But as the match wore on, they found a renewed sense of purpose.
Benteke carried their greatest threat and, having escaped the unstinting attentions of Toure, tested Mignolet with a curling shot that his fellow Belgian tipped around the post. Andreas Weimann's effort was then deflected just over the crossbar by Daniel Agger.
Villa began to break free of Liverpool's dominance in midfield and Agbonlahor began to flicker into life. The forward fired narrowly wide, while Benteke was denied a penalty as he collided with Toure following a corner. As with their opening-day win, Liverpool had failed to find a second goal and were now inviting pressure.
Benteke latched onto substitute Nicklas Helenius' header and drew another diving save from the excellent Mignolet. Liverpool might have won a late penalty as Guzan appeared to bring Sturridge down in the area but were happy enough to leave Villa Park with the points.