This victory, the Reds' fifth consecutive league win, secured with the additional help of a Steve Cook own-goal, extended Jurgen Klopp's side’s unbeaten start to the league season.
It also moved them one point ahead of City at the top of the table, hours before Pep Guardiola's side take on Chelsea at Stamford Bridge.
Salah's contribution was central to Liverpool's success – the Egyptian has struggled to match his outstanding levels of last season but with 10 league goals to his name, he is now hitting his stride.
He may have benefited from a close offside decision for the 25th-minute opener that gave Liverpool a half-time lead, but the finishes for his two second-half goals were outstanding.
Overall, Liverpool manager Klopp could be satisfied with the manner in which his side eased their way to victory ahead of the midweek Champions League meeting with Napoli.
Liverpool had demonstrated resilience to win on Wednesday against a physical Burnley side whose approach drew an angry reaction from Klopp.
Joe Gomez was a casualty of the trip to Turf Moor, suffering a broken leg in a challenge with Clarets defender Ben Mee, and with a testing week ahead Klopp named Trent Alexander-Arnold among the substitutes and named James Milner at right-back.
Bournemouth's hopes of halting Liverpool’s progress suffered a blow when Callum Wilson, the England international who has scored eight league goals this season, was ruled out with a hamstring injury.
The home side certainly felt Wilson's absence during the periods of the game when they pushed Liverpool onto the back foot.
But defensive errors, combined with a dubious offside decision for the opening goal, meant they had little chance to impose themselves on a Liverpool side that never had to find top gear to secure three points.
The opening period suggested Klopp's side would again have to work hard for victory. Salah and Roberto Firmino took time to make an impact, with the Egyptian spurning the first real chance of the game when he sliced wide from close range.
And had David Brooks shown more composure when he was sent clear on Liverpool's goal in the 24th minute, the early momentum would have been in favour of the home side.
As it was, Liverpool responded immediately, taking control with some help from the assistant referee.
Salah and Firmino combined well 25 metres out, before the Brazilian fired in a shot at keeper Asmir Begovic. The Bosnia international should have done better than parry the ball into the middle of the penalty area, allowing Salah to react first and fire high into the net from close range.
Eddie Howe's players pointed out that Salah had been in an offside position but their complaints were dismissed, although TV replays suggested they had a case.
From that point in the game, Liverpool tightened their grip, particularly after the restart.
It helped the Reds' cause that Salah was gifted the chance to add a second three minutes after the restart when Steve Cook lost the ball near the halfway line. Firmino released the Egyptian, who ran 40 metres before placing the ball beyond Begovic.
A bad afternoon for Cook got worse when he diverted the ball past Begovic and into his own goal as he attempted to cut out a left-wing cross from Andy Robertson.
And Salah produced the best goal of the game to wrap up the victory in the 77th minute after collecting Adam Lallana's through ball. The forward outpaced Cook and rounded Begovic twice to complete his treble.AFP