Salah set them on their way in the 25th minute, latching on to the rebound after Roberto Firmino's 30-metre shot was palmed into his path by Asmir Begovic, even though television replays showed the Egyptian to be marginally offside when he pounced.
There was no argument about his second straight after the interval when Firmino again freed him to maraud on goal and fire into the corner, with the help of the slightest nick off Nathan Ake's heel.
After the unfortunate Steve Cook diverted Andy Robertson's cross past Begovic for an own goal, Salah worked his real magic, outmuscling Cook to get to Adam Lallana's through ball before dancing nimbly around Begovic before slotting home.
Salah's treble at the Vitality Stadium put him on 10 goals for the season, matching Arsenal's Pierre-Emerick Aubameyang at the top of the scoring charts that he dominated last season.
Against tough, seventh-placed opponents, Juergen Klopp's side went a 17th straight league match unbeaten, equalling their club record set 10 years ago, and emulated their 4-0 thrashing of the Cherries in the corresponding fixture last season.
It put Liverpool a point clear of Manchester City at the summit on 42 points, though Pep Guardiola's men have the chance to regain top spot later when they visit Chelsea.
It was a splendid afternoon for Liverpool's James Milner to celebrate becoming only the 13th player to make 500 top-flight appearances in the Premier League era. (Reporting by Ian Chadband; editing by Ken Ferris)
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