The first 75 minutes were close with the Italians probably the better side as neither side managed a shot on target despite it being an open and entertaining affair.
But Firmino’s header opened the scoring and Salah then doubled the lead to give Jurgen Klopp a decisive lead ahead of the second leg at Anfield.
Liverpool’s high pressing caused problems for Inter early on as they regularly won the ball in dangerous positions.
Salah had a well blocked shot in the box, while Sadio Mane should have done better from a free kick as he headed over the bar.
Inter’s best moment of the half came when the ball was passed to Hakan Calhanoglu in the box and he sent an effort crashing against the bar, while Milan Skriniar headed just wide. a corner.
The home side also had some promising moments at the break, but a combination of poor final ball and brilliant defending from Virgil van Dijk and Ibrahima Konate prevented them from creating any clearer chances.
It was a brilliant start to Inter’s second half, with Marcelo Brozovic firing a shot wide and Ivan Perisic then just inches from laying it on a plate for Lautaro Martinez, but the cross simply had too much pace.
Perisic had a shot blocked before Edin Dzeko thought he had put the home side ahead, only to be quickly and correctly denied by the offside flag.
Klopp made a triple substitution in an attempt to regain control, with Jordan Henderson, Luis Diaz and Naby Keita all introduced. Diaz was brilliant for the Reds, having an effort blocked then whipping another from wide just wide.
Denzel Dumfries crossed the bar from a corner at the near post, as Inter continued to press and cause problems, but they couldn’t find a crucial goal.
They were made to regret it soon after, when Andy Robertson whipped in a corner at the near post and Firmino ran through his man to send a header into the far corner.
Inter looked devastated by the goal and things got worse for them with seven minutes remaining. Trent Alexander-Arnold’s cross was not cleared, it fell on Salah in the box and his shot went off Samir Handanovic on the wrong foot to find the back of the net.
The home side failed to find a late goal to give them something to bring to Anfield as Liverpool took a big step towards the Champions League quarter-finals.