Harry Kane gave Spurs a quick-fire lead in the rematch of June’s Champions League final when he planted in a header on the rebound within a minute of kick-off.
The Reds were on top for much of the first half without producing an equaliser, which instead eventually came via Jordan Henderson’s neat strike on 52 minutes.
And Salah settled a high-quality affair with a quarter of an hour remaining, arrowing in a penalty after Serge Aurier felled Sadio Mane in the box.
The victory put Jürgen Klopp’s side six points clear at the Premier League summit again and extended their unbeaten start to 10 games.