The Cherries, who contrived to lose 1-0 against 10-man Crystal Palace on Tuesday, have now lost five top-flight matches in a row and look set to be involved in a battle to avoid relegation in the second half of the season.
Jurgen Klopp made seven changes to the side that thrashed Everton 5-2 on Wednesday and Liverpool made the brighter start, with Mo Salah having an early shot saved by Aaron Ramsdale.
In the 25th minute, James Milner was only just off target with a fine strike before Eddie Howe’s team suffered a blow when Nathan Ake had to be substituted after injuring his hamstring.
It proved to be a turning point in the match as Liverpool took the lead almost immediately after Ake limped out of the action when Alex Oxlade-Chamberlain latched on to Jordan Henderson’s pass and touched the ball past Ramsdale.
After an unmarked Roberto Firmino had seen his shot fly wide, Liverpool doubled their lead just before half-time when Salah back-heeled the ball to Naby Keita who had the simple task of poking the ball home.
Klopp’s team continued to look dangerous after the break and after Keita had gone close to scoring his second goal of the afternoon, Salah made it 3-0 in the 54th minute after Jack Simpson had lost possession to Keita.
The Egypt international almost scored another goal with 20 minutes remaining but, perhaps with one eye on the vital Champions League match against RB Salzburg, Liverpool cruised through the rest of the game.