After a competitive start at Anfield, Egypt forward Salah curled high into the net from 18 yards, punishing Dusan Tadic for losing possession near his own area.
The strike gave the home side added swagger and a fine Philippe Coutinho threaded pass picked out an expertly-timed run as Salah prodded the second.
No Premier League player has more goals in all competitions than Salah’s 14 this season and though he was quieter after the break, Coutinho picked up the mantle to stab home when Roberto Firmino’s shot was saved.
In all, Liverpool had 21 shots at goal and are the only side to find the net three times against Mauricio Pellegrino’s team this season.
“It was an open game. We had more chances before we scored and we needed to be patient in a lively way,” said Klopp, whose side are fifth, a point behind Tottenham in fourth.
“The first goal – wow. I have nothing to do with the goal and the second was a fantastic pass and Mo Salah makes great runs. We could have scored more often but it was a perfect day for us.”