Salah's eighth top-flight goal of the season was allowed to stand by the officials to break the deadlock in a game the visitors dominated for long spells.
There was nothing fortuitous about the Liverpool forward's second though, a composed finish after driving at the Bournemouth defence, while Steve Cook back-heeled the ball into his own net while attempting a clearance to make it 3-0.
Salah then completed his hat-trick after rounding Begovic, cutting back to beat the Bournemouth keeper for a second time, and poking the ball past Nathan Ake on the line.
Having taken 45 points from the last 51 on offer, Liverpool are now unbeaten in 17 league games in a run stretching back to last season.
After a fifth straight league win, Klopp's side are one point clear of City, but the reigning champions were expected to return to the summit if they could avoid defeat at fourth-placed Chelsea later yesterday. AGENCIES