Philippe Coutinho opened the scoring for Liverpool in the 20th minute after some good work by full-back Andrew Robertson, who found the Brazilian who beat two defenders, before rifling a low strike past the helpless Asmir Begovic for a 1-0 lead.
It was soon 2-0 as Liverpool capitalised from a corner, as Lovren benefited from the sheer endeavour of Roberto Firmino, who did exceptionally well to keep the ball alive when it looked like a lost cause. The Reds frontman hooked the ball back into a dangerous area, to which Lovren reacted quickest to head home from close range.
The Cherries had a great chance to half the deficit when Jermaine Defoe beat the offside trap and Reds goalkeeper Simon Mignolet but the upright denied the former England international.
Mohamed Salah scored his 20th goal in a Liverpool jersey as the inform attacker produced a sensational piece skill to deceive Bournemouth defenders and sent a left-footed effort into the far corner.
The home side were unable to muster any real chances in the attacking third of the pitch while Mignolet was hardly tested in the second half.
Roberto Firmino capped off convincing display on the south coast when he nodded in a cross from compatriot Coutinho for an unassailable 4-0 with 15 minutes from the full time whistle.