While Manchester United relieved the pressure on manager Jose Mourinho with a 4-1 home romp against bottom club Fulham and Arsenal left it late to beat Huddersfield Town 1-0, Salah's best display of the season proved the day's headline act.
The livewire Egyptian, who had suffered a slow start to the season after his golden boot exploits of 2017-18, struck with what looked an offside rebound in the first half before scoring two individual beauties after the break.
A Steve Cook own goal for Liverpool's third summed up Bournemouth's misery at the Vitality Stadium as the Reds took their unbeaten streak to a club record-equalling 17 top-flight matches.
It put them on 42 points atop the table, one ahead of Manchester City, although the champions, also unbeaten this term, have the chance to go back to the summit when they visit Chelsea later on Saturday.
Juergen Klopp hailed his side's performance as "mature" and saluted the "absolutely brilliant" Salah as last season's golden boot winner clambered alongside Arsenal's Pierre-Emerick Aubameyang at the top of the goalscoring charts on 10.
For his part, Salah was determined to hand his man-of-the-match award to team mate James Milner, who felt this was the "perfect" way to celebrate joining the elite club of players who have made 500 Premier League appearances.
"I have to congratulate him on an amazing career -- he deserves this today," Salah said.
The grumbles around the stodgy form of Manchester United, who had not won in four league matches, were silenced at Old Trafford when goals from Ashley Young, Juan Mata, Romelu Lukaku and Marcus Rashford wrapped up an easy win over 10-man Fulham.
The visitors had Andre-Frank Zambo Anguissa sent off in the second half, minutes after an Aboubakar Kamara goal from the penalty spot had given Claudio Ranieri's side a glimpse of an escape.
"The fans today enjoyed it. That is the Manchester United that we want to see," said a delighted Mata.
Defeat left Fulham on nine points, alongside Southampton, who offered new manager Ralph Hasenhuettl a glimpse of the task he faces with their 1-0 loss at fellow relegation contenders Cardiff City, Callum Paterson scoring the 74th-minute winner.
Arsenal could not break down stubborn Huddersfield until Lucas Torreira, who also struck in last week's derby win over Tottenham Hotspur, built on his new cult status with a magnificent 83rd-minute bicycle kick winner.
It extended Arsenal's unbeaten run in all competitions to 21 matches and moved them, provisionally, into third place on 34 points, one clear of Spurs, who were playing at Leicester City in the day's late game.
Huddersfield's defeat left them on 10 points in 18th place in the relegation zone as Burnley leapt out of the bottom three with James Tarkowski's first-half goal earning them a 1-0 win over Brighton & Hove Albion at Turf Moor.
Most exciting match of the day came at the London Stadium where West Ham United came from an early goal down to beat Crystal Palace 3-2 with goals from Robert Snodgrass, Javier Hernandez and Felipe Anderson. (Reporting by Ian Chadband, Editing by Ed Osmond)
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