Salah took advantage of statuesque defending after 11 minutes to turn into the area and fire past Hugo Lloris, before Eric Dier miscued a headed clearance back to his goalkeeper to allow the forward to run clear and dink in a second.
The hosts had their moments with Ivan Perisic hitting the woodwork in each half and they could have had a penalty when Ryan Sessegnon was bundled over by Trent Alexander-Arnold.
After the break, Spurs raised their game, boosted by Dejan Kulusevski returning from injury off the bench. He had an almost instant impact as he fed Harry Kane who scored in the 70th.
But Liverpool weathered the storm to claim the win and end a run of two Premier League defeats, leaving them eighth on 19 points after 13 games. Spurs slipped to fourth on 26 from 14.
Additional reporting by Reuters.