The teams’ first ever league meeting was settled by £32.5 million ($50.8 million, 45.8 million euros) striker Benteke’s 26th-minute strike in front of the Kop, but Bournemouth felt that in attempting to play the ball before it reached his team-mate, Philippe Coutinho had been offside.
Having previously won 1-0 at Stoke City in their opening game, Brendan Rodgers’s Liverpool moved level on points with Manchester City, Leicester City and Manchester United at the top of the two-game table.
“Thankfully the goal counted and we’re happy to get the win,” said Rodgers.
“I’m pleased we’ve got two wins, six points and two clean sheets. The defensive organisation has been good. There’s a lot of players coming into the club and it’ll take them a wee bit of time to bed in.”
As well as Benteke, who later hit the crossbar, Rodgers also gave home debuts to Nathaniel Clyne, James Milner, Joe Gomez and substitute Roberto Firmino, but lost new captain Jordan Henderson to a foot injury early in the second half.
Bournemouth have lost both of their first games, but manager Eddie Howe will have been heartened by the pluck and enterprise shown by his side, who also had a goal by captain Tommy Elphick disallowed.
“It is hugely frustrating for us,” Howe told Sky Sports. “We were in the game and the big decisions have gone against us.
“The new offside rule is a grey area. Under the new rules, that is clearly offside. Our goalkeeper was affected by the player in front of him.”
The last time Bournemouth had played at Anfield, in an FA Cup replay in January 1968, it was under the banner of Bournemouth and Boscombe Athletic, but they were quick to make themselves at home.
It was the away fans who made all the noise in the early stages and in the fifth minute they thought they had a first ever top-flight goal to celebrate, only for Elphick’s header to be ruled out after he was adjudged to have used Liverpool centre-back Dejan Lovren for leverage.