Champions Manchester United will host Reading and Birmingham will visit Chelsea, while Arsenal take on Fulham.
And newly-promoted Sunderland scored a 1-0 victory over big-spending visitors Tottenham.
Blackburn earned a 2-1 win at Middlesbrough, hosts Derby drew 2-2 with Portsmouth, and Everton beat visitors Wigan 2-1.
Liverpool took the lead against Villa through a 31st-minute own goal by Martin Laursen.
The Villains equalised five minutes from time with a Gareth Barry penalty.
However, two minutes later, Steven Gerrard, the Reds captain, lashed in a splendid free kick.
Liverpool's next Premiership game is against Chelsea on August 19.
Gerrard said: "The idea this time is to come out of the blocks quickly and stay in the title race ... It was a great three points and nice to go into the Chelsea game with three points on the board."
At Upton Park, Eriksson got off to a flying start, with new strikers Roland Bianchi and Geovanni scoring in a stylish 2-0 win over the Hammers.
"I'm delighted, we played an excellent game," Eriksson said, singling out Brazil midfielder Elano. "We sometimes played the same system for England, but for England I never had an Elano.
"I don't know why it is but Brazilians seem to be born with technical ability in their DNA."
First choice striker
|Taylor dived, but missed Gerrard's |
spectacular free-kick [AFP]
Allardyce knows all about Bolton from his seven-and-a half years in charge at the Reebok and he put it to good use on his return in a 3-1 win.
Nigerian hitman Obefami Martins scored twice to show England forward Michael Owen he faces a fight to be the first choice striker.
Allardyce looked to bury his war of words with Phil Gartside, the Bolton chairman.
"I am very disappointed with the comments he has made," he said.
"But actions speak louder than words. We played them off the park and won the game. That answers everything for me, rather than get into a fight."
At the Stadium of Light, Michael Chopra scored the first goal of the new season in Saturday's early match as visitors Tottenham were beaten 1-0 thanks to his goal, scored 13 seconds from time.
Blackburn Rovers earned a hard-fought win over 2-1 Middlesbrough at the Riverside.
England winger Stewart Downing put Boro ahead, before South Africa striker Benni McCarthy was stretchered off after he collapsed following a clash of heads.
He was later released from hospital. His replacement, Paraguay striker Roque Santa Cruz, equalised and Matt Derbyshire scored a splendid winner.
Manchester City ended up with a 2-0 win
against West Ham [AFP]
Uefa Cup side Everton, out to improve on last season's sixth-placed finish, beat Wigan 2-1 at Goodison Park.
"We did the job and our play merited more goals than we did get," David Moyes, the Everton boss, said.
In a roller-coaster match at Pride Park, newly-promoted Derby County held visitors Portsmouth to a 2-2 draw.
Billy Davies, the Derby manager, said: "I am very proud of the performance. I know there is no one outside Derby who gives us a chance, and I am not surprised."
Pompey host Manchester United on Wednesday.
"I'd hate to play against them like that because they will rip us to pieces," disappointed Portsmouth boss Harry Redknapp said.