A contentious ninth-minute opener from Callum Wilson gave the Cherries an advantage that was wiped out by the leaders in the remainder of the first half.
Mohamed Salah marked his 100th league appearance for the Reds with the equaliser and Sadio Mane’s clinical finish later put them in front.
Jürgen Klopp’s side dominated the second half, too – but did require a pivotal James Milner clearance off the line – and did enough to post an all-time record 22nd consecutive home win in the English top flight.
Team news
Adrian continued between the posts due to Alisson Becker’s muscle injury, while Milner was deployed at left-back and Alex Oxlade-Chamberlain started in midfield.
Andy Robertson was omitted from the matchday squad as a precaution.
First half
Liverpool set about their customary brisk start in the lunchtime kick-off but it was the visitors who broke through first – despite the hosts’ protestations.
Wilson’s push on Joe Gomez was not deemed a foul as Bournemouth countered and they capitalised fully, the ball moving to Philip Billing, Jefferson Lerma and back into the middle for Wilson to convert from close range.
It was nearly two soon after as Adrian pushed Nathan Ake’s firm header from a right-wing corner onto the crossbar.
Liverpool reasserted themselves, though, and Roberto Firmino drew an important save from Aaron Ramsdale when he held off Steve Cook to collect Trent Alexander-Arnold’s prodded pass into the area and fired at goal.
The equaliser instead arrived on 24 minutes.
Mane spotted an opportunity to press Jack Simpson, recently introduced as a substitute, high up the pitch and successfully stole the ball and bore down on goal.
He elected to tee up Salah to his right and despite the pass coming slightly behind him, the Egyptian set his shooting position and threaded a sweet reverse finish into the bottom corner.
Simpson then had to clear from inside the six-yard box as Alexander-Arnold smacked a cross towards the waiting Mane.
The Reds did complete a turnaround before the interval, however, and this time Mane took care of the scoring himself.
Virgil van Dijk’s anticipation made the chance, the centre-back seizing on a loose ball around the halfway line and immediately sliding an assist into the run of an unmarked Mane. An unerring, swept shot beyond Ramsdale followed.
Second half
Liverpool dominated possession from the restart without generating any clear-cut chances.
And they required an outstanding piece of defensive recovery from Milner on the hour mark to preserve their slender lead.
Ryan Fraser neatly timed his run to break in behind the Reds defence off the flank and with Adrian heading out from his goalline, the Bournemouth man lifted the ball up and over him towards the empty net.
Milner rushed back, though, and stretched to produce a crucial clearance away from danger.
Van Dijk guided a header from a corner too close to Ramsdale and Firmino’s persistence then forced an opening, his driven cross-shot drifting wide of the target and Mane at the back post.
Next, Mane opted for a more direct route by dispatching a sumptuous curler from the left corner of the Cherries box that thudded off the upright and away, before Ramsdale touched Salah’s low hit wide of the same post.
In the end, a third was not required for a win that moves Liverpool within nine points of clinching the Premier League title.