It was the Egyptian who had the first chance of the game after seven minutes, receiving a cutback from Joe Gomez, but his curling effort was saved by Aaron Ramsdale.
Dominic Solanke had a point to prove against his former club and he nearly proved his point, only to see his goal-bound effort charged down by Dejan Lovren.
Up the other end, James Milner came within inches of opening the scoring only to see his effort from 15 yards flash past the far post.
Bournemouth's gameplan had worked a treat for the best part of half an hour until Nathan Ake sustained an injury. Liverpool capitalised on the Cherries' brief lapse in concentration.
It was a long ball over the top from Jordan Henderson to find Alex Oxlade-Chamberlain and, with Ramsdale reluctant to come off his line, he had to time to control the 70-yard pass before squeezing it underneath the goalkeeper.
Liverpool were rocked by an injury to Lovren with Trent Alexander-Arnold coming on but if anything, the change appeared to strengthen them. Indeed, when Salah cut inside and backheeled into the path of Naby Keita, he made no mistake to double the lead just a minute before half time.
Barely out of second gear, the Reds killed the game off just nine minutes after the break. Keita returned the favour this time, sliding through a sublime pass to cut the Bournemouth defence in half and Salah finished expertly past Ramsdale.
The home side endured more frustration when Callum Wilson was forced off with an injury and did have the ball in the back of the net, but Ryan Fraser was offside when he crossed to Arnaut Danjuma to score.
Salah and Alexander-Arnold both had chances to pile more misery on the hapless home side but they settled for a three-goal margin as Klopp's side widened the gap to 11 points at the top.