Liverpool should have won by a clearer margin in Tuesday's clash at Anfield, but Salah's 34th-minute strike was enough for the Premier League side to progress and for Napoli to be demoted down to the Europa League.
Both teams looked to get onto the front foot soon after kickoff and the first opportunity went to Salah, but the Egyptian failed to control the ball around 12 yards out after being found by Andrew Robertson.
Napoli went up the other end and Marek Hamsik fired over the crossbar from the edge of the area, before James Milner headed over for Liverpool as the two sides pushed for an opener in a game where Liverpool needed to either win with a clean sheet or by at least a two-goal margin.
Play continued at a fast pace without either team being able to create any clear-cut chances, although Liverpool did have a goal chalked off when Sadio Mane was correctly adjudged to be offside.
The tension around Anfield was only increasing as the minutes ticked by, but Liverpool went in front with 11 minutes of the first half remaining as Salah turned Mario Rui before drilling a shot inside the near post from an acute angle after Kalidou Koulibaly had failed to get close enough to the ball.
Napoli wasted little time in going back onto the offensive in search of an equaliser, but Fabian Ruiz wasted a good chance on the left-hand side of the area as he dragged a left-footed attempt well wide of the far post.
Liverpool needed to find a balance between attack and defence, but the home side showed more ambition after the restart with Salah clipping the outside of the near post with an effort from 14 yards.
Napoli still posed a threat at the other end with Raul Albiol failing to make the most of a header from a corner, but it was Liverpool who remained on the front foot with Jordan Henderson seeing a volley comfortably saved by David Ospina.
Milner then curled marginally wide of the far post but Napoli reminded their hosts that they were still in the game when Piotr Zielinski was able to get a shot away from inside the area, only for it to go over the crossbar.
Liverpool still went looking for a second goal, but after Salah and Mane were both denied through good work from Ospina, Virgil van Dijk sliced a volley over the crossbar having being found 10 yards out.
Napoli remained a threat on the break and Jose Callejon should have equalised in the 79th minute, but the Spaniard fired over from close range after a mistake from Robertson.
Liverpool could have made sure of the three points - and their progress - but more openings continued to be wasted with Mane firing wide from just eight yards out when a shot on target would have almost certainly led to a goal.
Four minutes of added-on time were signalled and Napoli almost netted the most dramatic of equalisers with Arkadiusz Milik being found on his own at the back post, but Alisson stayed big to make a stunning block at point-blank range.
Mane squandered a one-on-one opportunity with 60 seconds left but that miss - and the numerous ones before that - did not deny Liverpool another famous triumph on a European night at Anfield.
LIVERPOOL (4-3-3): Alisson; Alexander-Arnold (Lovren 90'), Matip, Van Dijk, Robertson; Wijnaldum, Henderson, Milner (Fabinho 85'); Salah, Firmino (Keita 79'), Mane
NAPOLI (4-4-1-1): Ospina; Maksimovic, Albiol, Koulibaly, Rui (Ghoulam 70'); Callejon, Allan, Hamsik, Ruiz (Zielinski 62'); Insigne; Mertens (Milik 67')