Roberto Firmino scored twice, while Darwin Nunez and Harvey Elliott also struck in a confidence-boosting win for Jurgen Klopp's men as they moved to within a point of qualifying for the knockout stages.
Liverpool had been forced to come from behind after Scott Arfield fired the Glasgow giants into an early lead.
But Rangers are being schooled on their return to the Champions League for the first time in 12 years as they remain without a point and have conceded 16 goals in four games.
Firmino's double on either side of halftime turned the game around before the Brazilian teed up Nunez to kill off the game as a contest.
68' - Salah comes on ➡️— Liverpool FC (@LFC) October 12, 2022
76' - Salah scores ⚽️
80' - Salah scores ⚽️
81' - Salah scores ⚽️
The Egyptian King ?
Klopp had left Salah and Thiago Alcantara on the bench with one eye on Sunday's clash against Manchester City in the Premier League.
But both were brought on for the final 22 minutes and Salah did not miss the chance to cash in against a beleaguered Rangers defence.
The Egyptian's treble arrived in just six minutes to beat the previous record held by Bafetimbi Gomis in Lyon's 7-1 win over Dinamo Zagreb in 2011.
Elliott then rounded off the scoring on a sobering night for a Rangers side that only months ago reached the Europa League final.
A draw in Amsterdam will suffice for Liverpool to reach the last 16 when they face Ajax in two weeks' time.
But Napoli remain out in front in Group A after a 4-2 win over Ajax sealed their place in the knockout stages.