Salah opened the scoring midway through the first half before adding two fine solo efforts to his tally either side of a spectacular Steve Cook own goal as Liverpool made it 16 games unbeaten from the start of a Premier League season for the first time in their history.
The Reds have now picked up a club-record 42 points from their opening 16 games to leapfrog Manchester City at the top of the table, at least until the champions face Chelsea at Stamford Bridge this evening.
The hosts were without injured top-scorer Callum Wilson, whereas Liverpool were able to welcome the likes of Salah, Roberto Firmino and Andrew Robertson back into their starting lineup, while James Milner made his 500th Premier League appearance.
Salah was the first to threaten as Liverpool settled quickly into the blustery conditions on the south coast, latching on to Xherdan Shaqiri's chipped ball forward but then slicing his right-footed effort wildly off target.
It took until midway through the first half for Bournemouth to begin to grow into the game, with Joshua King flashing one low effort across the face of goal before David Brooks drew a smart stop from Alisson Becker and Andrew Surman fired off target from the resulting corner.
Liverpool struck just when the hosts were enjoying their best spell, though, as Asmir Begovic failed to hold on to Firmino's long-range shot and parried the ball out into a dangerous area, where Salah was waiting to tuck home the rebound.
The goal saw Salah become the quickest player to reach 40 Premier League goals for the Reds - doing so in just 52 games - although replays showed that the Egyptian had just strayed offside when Firmino released his initial shot.
Bournemouth responded well to going behind without creating much to trouble Alisson, whose most taxing bit of work between the goal and half time came when he needed to race off his line and head the ball clear after Milner's sliced clearance swirled towards his own goal.
Liverpool struggled to conjure up many chances of their own before half time either, but they doubled their advantage within three minutes of the restart when Firmino released Salah, who stayed up despite a raking challenge down his Achilles and then proceeded to pick out the bottom corner with a measured finish from the edge of the box.
The hosts almost halved the deficit just four minutes later when King beat Alisson to a low cross from the left, but he went down in search of a penalty when he may have been better served trying to engineer a shot from close range.
Liverpool's two-goal cushion saw them take complete control of the contest, though, and they added a third in fortuitous circumstances with just over 20 minutes remaining when Cook spectacularly backheeled the ball into his own net from Robertson's testing cross.
Bournemouth had only failed to score in one home league game all season before today, and they went in search of a consolation to maintain that record when Junior Stanislas curled a 30-yard free kick towards the top corner, only for Alisson to make his save look easy.
The fourth goal arrived with 13 minutes remaining as Salah saved the best of his hat-trick until last, beating Cook to the ball and then taking his time with the finish, dribbling past Begovic twice before nonchalantly poking the ball away from two defenders on the line.
In doing so, Salah drew level with Pierre-Emerick Aubameyang as the Premier League's top scorer this season, and he was given the chance to see out the game by Klopp despite matches against Napoli and Manchester United next week.
Liverpool's best chance to add a fifth fell to Georginio Wijnaldum, though, as he nodded Robertson's cross off target when he should have done better.
The points had long since been wrapped up, though, as Liverpool equalled their club record of 17 Premier League games unbeaten to pile the pressure on Man City ahead of their trip to Stamford Bridge later today.
The Reds are just the fourth team in the Premier League era to have begun a season by avoiding defeat in their opening 16 matches, joining Arsenal's Invincibles, Manchester United's 2010-11 team and Manchester City last term in achieving the rare feat.
Klopp's side will now face a must-win game against Napoli on Tuesday night before welcoming bitter rivals Manchester United to Anfield next Sunday.
BOURNEMOUTH (4-5-1): Begovic; Francis, S Cook, Ake, Daniels (Rico 83'); Stanislas (Mings 83'), Surman, Brooks (Mousset 65'), Lerma, Fraser; King
LIVERPOOL (4-3-3): Alisson; Milner, Matip, Van Dijk, Robertson; Shaqiri (Mane 65'), Fabinho, Wijnaldum, Keita (Lallana 65'); Firmino (Henderson 81'), Salah