Salah struck once before half-time and twice after it, with a Steve Cook own goal in between, as Jurgen Klopp's side recorded the most comfortable of victories.
After a slow start in which Liverpool dominated possession, the visitors carved out the first opening when Roberto Firmino's pass to Salah hit Nathan Ake's arm but Xherdan Shaqiri's free kick deflected to safety.
Ake was booked for handball and Liverpool came forward again as the 15-minute mark approached with Shaqiri finding Salah who, from an angle, lashed his strike into the crowd.
With Bournemouth struggling to put anything together in the south coast rain, Jefferson Lerma joined Ake in the book for a foul on Fabinho and an excellent Ake challenge ended Salah's threatening run.
Naby Keita had a shot blocked after loose defending inside the Bournemouth box before the hosts flickered into life when a determined run by Josh King took him to the edge of the Liverpool area but he scuffed wide.
Andrew Surman then threatened with a fizzing shot from distance but, within seconds, Liverpool raced to the other end to make the breakthrough.
Salah appeared offside when Asmir Begovic could only block Firmino's effort from the edge of the box but followed up to fire past the keeper from 10 yards and put Klopp's side in front.
With Bournemouth on the back foot, an excellent tackle from Charlie Daniels prevented James Milner from getting clear down the right, but the home side rallied to win a corner from which a Junior Stanislas strike was deflected wide.
Home forward Josh King collided heavily with Liverpool keeper Alisson as the two contested a high ball inside the penalty area but was able to continue after treatment as half-time came with Liverpool in the lead.
Within moments of the second half getting under way, the lead was doubled when Firmino fed Salah midway inside the Bournemouth half and he ran into the area to slide a low finish beyond Begovic.
Daniels found King inside the area as Bournemouth looked for a way back into the game but the forward, under pressure from Alisson, lifted wide.
Lerma's attempt from outside the area was blocked by Keita before Francis made a determined run to the edge of the area only to roll his final pass away from waiting teammates.
With 65 minutes gone and an eye on Tuesday's crucial Champions League clash against Napoli, Klopp replaced Sahqiri with Sadio Mane and Keita with Adam Lallana, and soon saw his side take a 3-0 lead when Andy Robertson's cross cannoned in off the unfortunate Cook.
Fabinho was booked for a clattering foul on Lerma 25 yards out, and Stanislas forced Alisson into action with a curling free kick before King lashed home from a Surman pass but was thwarted by an offside flag.
Salah had the final word with 13 minutes remaining, running in behind Cook and dancing round Begovic before steering a nonchalant finish away from two Bournemouth defenders on the line.
In a flurry of late changes, Liverpool replaced Firmino with Jordan Henderson before Bournemouth brought on Tyrone Mings for Stanislas and Diego Rico for Daniels, who had come close with a fierce 25-yard drive moments earlier.
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