The Egyptian came off the bench to add Liverpool's fourth, fifth and sixth goals of the evening at Ibrox as Jurgen Klopp's side returned to winning ways with an emphatic 7-1 victory.
There were just six minutes and 12 seconds between Salah's first and third goals, making it the fastest hat-trick in Champions League history.
Salah also needed only nine touches to secure the matchball - the fewest taken by any player to have scored any of the 112 Champions League hat-tricks since full touch data became available in 2003-04.
In addition to that, Salah's second goal saw him move outright on his own as the player with most Champions League goals for a single English club.
The Liverpool forward had gone into the game sitting on 35 goals in the competition for the Reds, one behind the tallies that Didier Drogba and Sergio Aguero managed for Chelsea and Manchester City respectively.
However, he has now moved to 38 for Liverpool and 41 in total in the competition, equalling Aguero's overall haul and becoming only the 18th player to pass the 40-goal mark.
There was also a small slice of history for Diogo Jota, who became the first player to assist every goal in a Champions League hat-trick for a teammate since Franck Ribery did the same for Mario Gomez at Bayern Munich in March 2012.
Despite Salah's treble and Jota's hat-trick of assists, arguably the standout performer was Roberto Firmino, who netted his sixth Champions League brace for Liverpool and chalked up his 12th assist for the club in the competition, drawing level with Steven Gerrard and James Milner at the top of those charts for Liverpool.