Liverpool were struggling to create many opportunities, with coach Juergen Klopp ringing the changes in the second half, and one of those substitutes came up trumps - Firmino flicking home from a corner to break the deadlock in the 75th minute.
With their resistance broken, the errors started to creep in for Inter and Salah pounced when they failed to clear a long ball into the box to put the game to bed seven minutes from time.
Even with the away goals rule no longer in force, the victory puts the 2019 Champions League winners in a commanding position ahead of the return leg on March 8 at Anfield.
Meanwhile Bayern Munich midfielder Kingsley Coman struck in the 90th minute to snatch a 1-1 draw at Salzburg.
The Austrian side, who were competing in the knockout stage of the competition for the first time, took the lead midway through the first half with a goal on the counterattack from Chukwubuike Adamu.
Photo: PHOTOSPORT Caption: Roberto Firmino