The Egypt winger was slightly below par on this occasion by his lofty standards, but a header 21 minutes from time took him to 40 in all competitions in a Reds shirt and 30 in the league, adding to Sadio Mane's first-half strike before Roberto Firmino netted a late third.
The Reds - inspired by Salah - booked their place in the Champions League semi-finals with a win at Manchester City on Tuesday, but their attention returned to domestic action in a victory that puts pressure on second-placed rivals Manchester United.
Jurgen Klopp named a full-strength starting XI and it quickly became clear that Bournemouth would not improve on their winless record at Anfield.
It still looked for long periods as though Liverpool would fail to add to the lead secured early by Mane in a one-sided match, but Salah predictably sealed the win to move five clear of Harry Kane in the Golden Boot race ahead of the Tottenham man's clash with City later in the day.
And Firmino got in on the act in the final stages as Klopp's side move within a point of United, who have two games in hand.
Pre-match tributes were held on the eve of the 29th anniversary of the Hillsborough disaster, before Liverpool began on the front foot and Salah toed Trent Alexander-Arnold's pass from deep just wide of the right-hand post.
It took only seven minutes for the opener to arrive. Jordan Henderson crossed following a half-cleared corner and Mane flicked a header towards goal, then pounced to slam home the rebound when Asmir Begovic could only parry.
Routine Begovic saves from Alexander-Arnold and Salah followed, with the home side relentless in attack and their pace terrorising a scrambling Bournemouth back-line.
Further clear openings did not arrive before the break, though, and Bournemouth dealt with the dominant hosts rather more confidently after the restart.
Still there remained very little goal threat at the other end - with Jermain Defoe anonymous - allowing Salah time to finally find his range.
The winger looped a sublime header over Begovic to settle the contest and reach another scoring milestone, before teeing up Mane for a shot that skidded wide.
Firmino somehow failed to add his name to the scoresheet shortly afterwards when he rounded Begovic and could not finish, but he got his goal in the closing stages, squeezing a low effort in between goalkeeper and post.
Liverpool will now surpass a year unbeaten in home league games before Stoke City visit in a fortnight - by which time Roma will have been to town for a Champions League semi-final clash that has rather more riding on it than this game.
