Inter had the better of the first-half chances, with Hakan Calhanoglu firing against the crossbar when afforded too much time in the penalty area.
Liverpool were struggling to create many opportunities, with coach Jurgen 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Even with the away, goals rule no longer in force, the victory puts the 2019 Champions League winners in a commanding position ahead of the return leg on March 8 at Anfield.
Liverpool now turn attention to the Premier League title race as they face Norwich City at Anfield while Inter Milan face Sassuolo.