Victor Moses has put Liverpool on course to progress to the fifth round of the FA Cup, but Brendan Rodgers's men have come under concerted pressure from an impressive Bournemouth side.
The Dorset club, who came into tie on the back of only one defeat in their last seven games, started much the brighter and had an early chance when Andrew Surman's glancing header drifted wide of Brad Jones's goal.
Liverpool began indifferently as they struggled to put any pressure on the Bournemouth defence in the opening minutes.
The Cherries continued to enjoy the better of the game and almost took the lead when a free kick from Simon Francis needed keeper Jones to tip the ball over the crossbar.
Indeed, when Liverpool took the lead on 26 minutes thanks to Victor Moses's low drive that beat Lee Camp at his near post, the goal had come against the run of play.
Bournemouth continued to press the Reds defence, but didn't have the first clear shot at goal until just after the half-hour mark when Harry Arter's effort had Jones sprawling across his goal, but his effort drifted wide in the end.
Their two best chances of an enterprising first half came 10 minutes before the break. First, a ball cut back to the centre of the area found Surman unmarked, but his effort was diverted over, and from the following corner, Surman picked out the run of Tommy Elphick, whose diving header narrowly cleared the bar.
Moments later, Jordan Henderson ought to have put daylight between the sides when a rare Liverpool attack reached the England midfielder at the heart of the area, but his effort failed to trouble Camp.
With the interval approaching, Liverpool started to attack with greater freedom and control and Luis Suarez's jinking run and shot was seen off by a watchful Cherries defence.
Liverpool exited the FA Cup at the fourth round stage last year after a shock 3-2 defeat at Oldham Athletic.