The visitors made the breakthrough against the run of play on 26 minutes when Victor Moses fired into the bottom corner from the edge of the penalty area.
Here, Sports Mole takes a closer look at the action between the two teams at Dean Court.
On target: 3
On target: 4
Was the result fair?
Despite Bournemouth giving an excellent account of themselves, it was Liverpool who came out worthy winners thanks to their clinical finishing in front of goal. They made hard work of it at times, but Brendan Rodgers's side will be happy with their professional display.
The Cherries came flying out the traps with plenty of energy and commitment, forcing Liverpool to make mistakes during a decent first half for the hosts. They showed no fear going forward and would have had a goal but for some wayward finishing in the final third. The Championship outfit were able to keep it up for most of the second period, although they looked tired towards the end as the visitors gradually picked them off. Still, manager Eddie Howe is sure to be pleased with his players for their stamina and determination against high-class opposition.
Manager Brendan Rodgers, who chose a strong starting XI, will be the first to admit that his side played below par, but they got the job done in the end. Their first-half display was marked by a sloppiness as they struggled to find their usual rhythm, although they still managed to be ruthless in front of goal and take the lead through Victor Moses. The Merseysiders were much improved after the break and kept Bournemouth out of reach before killing the game off thanks to Daniel Sturridge. While this was an average performance overall, the Reds are at least in the hat for the last 16.
Sports Mole's man of the match
Harry Arter: The Bournemouth midfielder was in no way daunted by the world-class talent of Liverpool as he provided the hosts with creativity and a touch of finesse. He was able to link up well with Charlie Daniels on the left side and was at the heart of almost everything positive that the home side tried to do.
Liverpool had a great opportunity to make it 2-0 towards the end of the first half, but Jordan Henderson blazed over from around 12 yards after being fed by Luis Suarez. The midfielder really should have done better as he had time and space in the box.
Lee Probert did little to hamper the flow of an entertaining cup tie as he was competent throughout.
Bournemouth: The Cherries are back in action on Tuesday as Huddersfield Town visit Dean Court in the Championship.
Liverpool: The Reds continue their Premier League campaign on Tuesday when they welcome local rivals Everton to Anfield in the Premier League.