A year ago, the Reds were bundled out at this stage of the competition in a shock 3-2 reverse at League One outfit Oldham.
Skipper Steven Gerrard started that game on the bench, but he was named in the side on Saturday against Eddie Howe's Cherries, currently lying 16th in the Sky Bet Championship.
And despite a spirited display from the south-coast club, they were unable to provide an upset at the Goldsands Stadium.
On-loan Chelsea winger Victor Moses opened the scoring in the 26th minute with a low shot into the corner from the edge of the box and, after Bournemouth had had a second-half penalty appeal turned down, Daniel Sturridge was put through on the hour mark to double the visitors' lead.
Sturridge's composed finish will have pleased watching England manager Roy Hodgson, although the most influential player on the pitch with both assists was Luis Suarez, whose Uruguay side face the Three Lions at the World Cup in just under five months' time.
Liverpool manager Brendan Rodgers also included Brad Jones and Martin Kelly in his starting line-up, while Howe named the same XI that drew 1-1 at home to Watford last time out.
Jones had to be alert early on as Simon Francis' deep free-kick eluded his Cherries team-mates, with the ball pushed over by the Australian.
Marc Pugh then headed Matt Ritchie's free-kick wide at the back post, but it was the Reds who forged ahead just past the midway point of the first half.
Suarez got away down the right flank past Elliott Ward and picked out a fine looping pass right across the penalty area to Moses who was waiting on the left corner of the area.
The Nigeria winger cut onto his right foot and fired a low shot inside the near post and past the despairing dive of Lee Camp, who was perhaps unsighted by the defenders in front of him, but will be disappointed to have been beaten in such a fashion.
Harry Arter flashed a free-kick wide, Andrew Surman saw a shot deflected behind and Cherries captain Tommy Elphick headed over from the resulting corner.
However, it was Liverpool who came closest to scoring again before the interval, only for Jordan Henderson to fire over the bar after Suarez had wriggled free in the box.
Bournemouth continued to push for an equaliser and on 58 minutes, Kelly was fortunate not to concede a penalty when he pulled the shirt of Francis on a corner.
But shortly afterwards, Liverpool doubled their advantage. Suarez slid a perfect pass into the run of Sturridge, who coolly clipped the ball beyond the advancing Camp and inside the far post.
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 actually had more attempts on goal throughout the 90 minutes but with their 'SAS' partnership in such good form, Liverpool were able to see the game out and secure a spot in Sunday evening's draw for the last 16.
|0||1st Half Goals||1|
|3||Shots on Target||4|
|10||Shots off Target||6|
|45.7||1st Half Poss.||54.3|
|44.1||2nd Half Poss.||55.9|