The home side started brightly, with Callum Wilson missing a glorious chance to put them in front in the opening five minutes.
However, Liverpool waited for their moment, and a fine move – involving 52 passes – resulted in Sterling heading Brendan Rodgers’ side into the lead.
Lazar Markovic, who was at the heart of most of Liverpool’s best attacks, then scored his first goal for the club after Philippe Coutinho had seen his strike saved by Artur Boruc in the Bournemouth goal.
Sterling then scored his second shortly after half-time before Dan Gosling got Bournemouth on the scoresheet, sliding the ball under Brad Jones in the Reds’ goal – but the visitors held on for a result that eased some of the pressure on Rodgers.
As early as the fifth minute, Bournemouth top scorer Wilson should have broken the deadlock, the former Coventry striker rounding Martin Skrtel and Dejan Lovren, but then dragging his shot wide when through on goal.
At the other end, Boruc did brilliantly to tip a fierce Adam Lallana effort over the bar, as the Reds started to show their quality.
Then, after a long spell of possession, Liverpool opened up the Championship leaders after 20 minutes, Markovic looping a ball to the back post, where Jordan Henderson nodded across goal for the onrushing Raheem Sterling, who had the simple task of heading in from three yards.
Just seven minutes later, Liverpool were two up in controversial circumstances.
Markovic was again pivotal as he played Coutinho in but Tommy Elphick then stopped after hearing a whistle – which came from the crowd – allowing the Brazilian to cut inside and shoot.
The shot was saved by Boruc but the loose ball fell to Markovic on the edge of the box, and the Serbian placed his shot into the corner.
Bournemouth had a great chance almost straight from the restart but after being picked out in the box Yann Kermorgant fired over with the goal gaping.
Six minutes into the second half, Liverpool extended their lead, Henderson playing Sterling through, who twisted and turned before powerfully firing past Boruc.
Bournemouth pulled a goal back just before the hour, Matt Ritchie picking out Gosling on the edge of the box before the former Everton midfielder showed great composure to finish past Jones in the Liverpool goal.
Sterling spurned a great chance to net his hat-trick after being picked out by a fantastic Lallana flick, but the England forward got it all wrong, and side-footed wide from close range (57).
Liverpool briefly wobbled and were given a real scare when Gosling curled a strike against the post with Jones beaten, – but that was the last time the Liverpool keeper was troubled.
As the game wore on, the home side tired, and Liverpool did well to slow the game right down and see the game out.
Bournemouth: Boruc (7), Francis (7), Elphick (6), Cargill (6), Smith (7), Gosling (8), O’Kane (6), Ritchie (7), Kermorgant (5), Stanislas (6), Wilson (6)
Subs: Arter (6), Pitman (6), Fraser (7)
Liverpool: Jones (6), Toure (7), Skrtel (6), Lovren (6), Henderson (7), Gerrard (7), Lucas (7), Markovic (8), Coutinho (8), Sterling (9), Lallana (7)
Subs: Sakho (6), Can (6), Borini (5).
Man of the Match: Raheem Sterling