The £32 million summer signing from Aston Villa struck from close range after 26 minutes but debate raged over whether Philippe Coutinho was offside.
This came after the Cherries had earlier had a Tommy Elphick goal harshly ruled out for a push in the area by referee Craig Pawson. Bournemouth pressed hard in the second half, hitting the post through Scotland international Matt Ritchie, but Liverpool did enough to close out a second successive win.
Brendan Rodgers, with Liverpool owner John W Henry watching from the stands, fielded a side unchanged from the laboured opening victory at Stoke. They started in much the same sluggish manner as the visitors made the early running. Max Gradel looked dangerous on the left, Andrew Surman and Eunan O’Kane provided drive and strikers Callum Wilson and Josh King were threatening. The first moment of contention came in the fifth minute as Elphick was penalised for fouling Dejan Lovren as he manoeuvred into space to head home from a corner. It was a tough call but Bournemouth were not disheartened and continued to force the pace.
It was the 24th minute before Liverpool finally posed a serious threat, with Benteke driving narrowly over from distance.
The questionable goal came moments later as Jordan Henderson swung in a deep cross from the left.
Coutinho was offside and clearly played at the ball as it bounced past him on the way to Benteke, who turned home at the far post.
The goal, Benteke’s 50th in English club football, sparked strong debate over the latest interpretation of the offside rule.
Coutinho did not have to make contact to be flagged, as he would have done in previous seasons, but his action had to impact on an opponent.
There were no Bournemouth players near him as he moved for the ball, but goalkeeper Artur Boruc may have anticipated a touch. Again it was harsh on the visitors and they lost momentum after the setback.
Liverpool upped their tempo and Henderson clipped the top of the bar with a long-range shot and Coutinho fired just wide. Coutinho saw another effort deflected just off target early in the second half.
Bournemouth responded after Henderson, presumably injured, was taken off after 52 minutes.
Ritchie grew in influence on the right, smashing a shot against the woodwork, having another chance blocked by Joe Gomez and then seeing an effort deflected for a corner. Wilson teed up an opportunity for O’Kane, but he was only able to hit the side-netting from a tight angle.
Liverpool did enough to hold on and Benteke almost snatched a second in the closing moments when a deflected shot struck the bar.