Firmino came off the bench at San Siro and opened the scoring with a clinical header from a corner in the 75th minute.
And eight minutes later, Salah netted to give the Reds a two-goal advantage when they welcome the Serie A champions to Anfield in next month's return fixture.
Team news
There were four changes to the starting line-up that beat Burnley on the weekend. Ibrahima Konate, Thiago Alcantara and Diogo Jota came into the XI, while there was a Champions League debut for Harvey Elliott.
First half
There were openings at both ends of the pitch during an entertaining opening 45 minutes.
Lautaro Martinez and Thiago both went close with efforts from the edge of the box early on, before the clear-cut chances arrived.
Sadio Mane headed fractionally over from inside the penalty area after rising highest to meet Andy Robertson's free-kick. Then, Hakan Calhanoglu pounced to receive an Ivan Perisic cross from the left, striking the crossbar from a tight angle and watching the rebound bounce clear.
There was nearly a golden opportunity for either side following counter-attacks, though they were thwarted by vital interceptions from Calhanoglu and Konate.
And the final chance of the first half went to the hosts as Milan Skriniar headed fractionally wide from an out-swinging corner.
Second half
A tactical change for the Reds at the break saw Firmino replace Jota.
The hosts again came close to breaking the deadlock early in the second period. Perisic's delivery following Denzel Dumfries' switch was just out of the reach of Martinez in the six-yard box.
Under heavy pressure, Liverpool then opted for a triple change, bringing on Luis Diaz, Naby Keita, Jordan Henderson in place of Mane, Elliott and Fabinho.
Diaz, on his European debut for the club, nearly scored within minutes of his introduction, but saw a close-range shot stopped by a last-ditch block from Skriniar.
But the visitors made their next opportunities count.
To open the scoring, Robertson fired an in-swinging corner into the box, where Firmino met it and glanced a fine header into the far corner.
And to wrap up the victory, Alexander-Arnold pumped a free-kick into the area that was nodded down by Virgil van Dijk. In position to latch on to the loose ball, Salah hit a first-time strike towards goal that took a slight deflection off an Inter player en route to the back of the net.