Liverpool too hot for Bournemouth
The visitors named a strong side for the game at the Goldsands Stadium and it was Luis Suarez who created both goals to see the Reds into the last 16.
Cherries' boss Eddie Howe had said leading into the game that he hoped, even if his side lost, they would show people that they could play attractive football.
That was the case for much of the game as the Sky Bet Championship outfit created more chances than their Premier League opponents - but it was the clinical nature of Liverpool's finishing which ultimately proved the difference.
Bournemouth started the brighter of the two sides, forcing two early corners before Andrew Surman headed Charlie Daniels' cross over in the third minute.
Liverpool could not settle as the hosts refused to be overawed - playing in their traditional passing style as Howe had promised ahead of the match.
Brad Jones, playing in goal for Liverpool with Simon Mignolet rested, had to be alert as Simon Francis' deep free-kick eluded his team-mates but had to be pushed over by the Australian.
Marc Pugh then headed Matt Ritchie's free-kick wide at the back post before Liverpool finally sparked into life.
Suarez was fouled by Harry Arter but Steven Gerrard's effort from the resulting free-kick was high and wide as Lee Camp continued to go untested in the Bournemouth goal.
That changed seconds later as Suarez breezed past Elliott Ward and crossed to Moses - who arrowed a low shot past Camp's despairing dive to put the visitors ahead.
Bournemouth did not let their heads drop as Arter's well-struck effort flashed just wide with the Cherries going in search of an immediate reply.
Howe's side continued to create chances with Surman seeing a shot deflected behind and captain Tommy Elphick heading the corner over.
But it was Liverpool who spurned a glorious opportunity to double their lead as Suarez wriggled free in the box and found Jordan Henderson.
The midfielder was in space but could only put his shot well over the bar as the Reds looked to score a second smash and grab goal.
The start to the second half was fairly even with Bournemouth continuing to ask questions of their top-flight opponents.
But the ruthlessness of Liverpool's front players soon turned the tie fully in their favour on the hour-mark.
Suarez delivered a fantastic pass to strike-partner Sturridge, with the England international slotting past Camp with aplomb to score his third goal since returning from an ankle injury.
The former Chelsea forward came close to adding a third as he latched on to Henderson's long ball and lobbed Camp - with his effort going behind off the crossbar.
With the hosts chasing a way back into the game Sturridge once again worked some space but could only shoot wide after cutting inside.
Suarez was the next to come close as he beat the offside trap to attempt his own lob, sending his effort over goalkeeper and crossbar.
Bournemouth searched in vain for a late consolation as Brendan Rodgers' side took the sting out of the closing stages.