There had been frustration among supporters after successive draws with Sevilla and Chelsea, but the Reds responded in fine style at the bet365 Stadium.
The manager cut a typically laid-back figure in the early exchanges, as the game got off to a tight start.
It was the returning Sadio Mane who struck first, with the Senegalese dinking a lovely finish over Lee Grant and into the net.
Simon Mignolet could easily have been sent off before half-time, having hauled down Mame Biram Diouf, but he was only shown a yellow card.
The second half was all about Salah, who came on for Alex Oxlade-Chamberlain and rifled a stunning volley into the net to kill the game off.
The Egyptian then added a second late on, beating Grant after racing onto Emre Can‘s long ball.
Next up for Liverpool is a trip to Brighton on Saturday afternoon (3pm GMT) as they look to make it five Premier League wins out of six.