Ahead after seven minutes through Sadio Mane, the Reds completed the job via Salah's second-half header and a Roberto Firmino strike to edge closer to a confirmed place in the Champions League next season.
Early opener makes for comfortable win
Much has been made of Mane's supposed 'off-season', during which he's plundered more than in his 'fantastic' debut season on Merseyside, number 17 of this campaign coming against the Cherries.
Jordan Henderson did well to angle a good cross in as a corner was swept away by Eddie Howe's men, Mane's original header kept out by Asmir Begovic before the winger recovered quickly to slam a rebound past both the 'keeper and Steve Cook.
That set the tone for a comfortable afternoon for the Reds, Loris Karius not troubled by a shot on target in the first 45 minutes.
Salah gets his customary goal
Their dominance in possession should have led to more goals for the break, Trent Alexander-Arnold close with a free-kick whilst Nathan Ake did well to sniff out chances for Premier League top scorer Salah.
Liverpool's fans, in buoyant mood after their Champions League display against Manchester City on Tuesday, were made to be patient before being rewarded with Salah's 40th competitive goal of the season.
Alexander-Arnold was the provider, swinging a fantastic ball into the area which found Salah, only for the Egyptian to better that with a genius header that looped over Begovic and dropped into the net.
That goal was the trigger for Bournemouth's kitchen sink which, once dropped into the Liverpool half, gave Jurgen Klopp's men much more room on the counter.
Mane fired one wide in a bid to get his second of the afternoon, whilst Nathan Ake's goal-line clearance prevented Firmino from getting in on the act after he rounded the Begovic.
Bournemouth would chances of their own eventually as cross bundling off Callum Wilson before falling to Dan Gosling, who saw his shot well beaten out by Loris Karius.
Wilson would then get the chance to pull the trigger himself as a hopeful ball kindly fell beyond the home defence, the striker himself unable to keep his effort on target.
Reds notch one more
That was enough for Liverpool to wise back up and shut the door on any more Cherries chances, closing the game out themselves with a goal for Firmino.
Alexander-Arnold started the move by finding Wijnaldum, who in-turn let Alex Oxlade-Chamberlain drive forward and slip the ball to the Brazilian, who checked back inside and fired through Cook's legs to beat Begovic at his near post.