Jurgen Klopp found his side 1-0 down early on when Scott Arfield guided a ball past Alisson from outside of the box, but Roberto Firmino led a quick response from Liverpool to equalise with an accurate header from a corner.
An even first half saw little else created by way of chances, but Rangers had the best one when Fashion Sakala flashed past Kostas Tsimikas before cutting the ball across the goal, only for Ibrahima Konate to deny Arfield from securing his brace.
Liverpool looked to have much more control in the second half as they stemmed the flow of attacks from the opposition, and the Premier League side got themselves into the lead when Joe Gomez linked up with Harvey Elliott before delivering a precise low cross into the path of Firmino, who made no mistake with the finish.
The Reds maintained their pressure after getting in front and found a third as Darwin Nunez finished well from Firmino's flick, and they were soon out of sight after Mohamed Salah was introduced. The right-winger scored three times in six minutes after missing his first chance, with Harvey Elliott scoring the first Champions League goal of his career for Liverpool's seventh.
An excellent second-half performance saw Liverpool restore their lethal edge. What turned into a dominant performance netted a positive result in spite of injury problems ahead of the trip to Manchester City on Sunday.
The Reds were slow to start the game with a poor first half where they continued their recent trend of conceding the first goal.
Manager rating (out of 10 with 10 = best)
Jurgen Klopp, 8 -- Klopp had a number of injury problems coming into this one and managed to deploy a system that suited his line up. His substitutions were timed well to ensure a comfortable victory in the end.
Player ratings (players introduced after 70 minutes = no rating)
GK Alisson, 6 -- There wasn't much to do for Alisson outside of the goal he conceded as Liverpool's defence prevented Rangers from getting any further dangerous efforts on target. Reacted quick to Konate's block that stopped a cross reaching its intended target in the second half.
DF Konstantinos Tsimikas, 6 -- A strong delivery saw Tsimikas register an assist for Firmino's equaliser, but the Greece international could have been better with some of his defending in isolated situations on the night.
DF Virgil van Dijk, 7 -- He took some time to get into his best gear but looked comfortable once he did, anticipating danger quickly and showing calmness under pressure in possession.
DF Ibrahima Konaté, 8 -- The Liverpool centre-back was often the man to clear the danger, and got in the way of a strike inside the box close to half-time after Sakala cut the ball across the danger area. A stellar display from the France international.
DF Joe Gomez, 7 -- The 25-year-old kept wide in moments when he marauded forward, and he produced an exquisite pass in the second half that found Firmino for Liverpool's second goal of the night.
MF Fabinho, 6 -- Anchored the defence to cut-off spaces and that was vital in Liverpool's third goal as Fabinho won back possession high up the pitch and progressed the ball forwards quickly.
MF Harvey Elliott, 8 -- The 19-year-old was one of Liverpool's brighter players in possession throughout the game, and looked willing to take risks with progressive play. Carried the ball forward when required to link the midfield to the attack. A well-deserved goal to cap off his performance.
MF Jordan Henderson, 6 -- Progressed the ball safely and provided cover alongside Fabinho. Helped organise Liverpool's midfield after a slow start in the first half.
MF Fabio Carvalho, 7 -- A mixed first half saw the play break down on a number of occasions through the young attacking midfielder, though he did look better when playing from central areas.
FW Darwin Nunez, 7 -- He made positive runs into space that helped create opportunities for Liverpool, and he got on the scoresheet with a well-taken finish when combining with Firmino.
FW Roberto Firmino, 9 -- Another impressive display from Firmino saw him score his eighth goal in nine games with a brace on the night against Rangers. He also picked out Nunez with some flair to assist Liverpool's third.
Mohamed Salah (Nunez, 68"), 10 -- The winger was dangerous when introduced and while he missed his first chance, he made no mistake with his next three as he completed a hat-trick in five minutes.
Thiago (Henderson, 68"), 7 -- Thiago was aggressive in the challenge and always looked to play the ball forwards. Won possession back for Salah's second goal.
Andy Robertson (Tsimikas, 68"), 6 -- Brought on for Tsimikas and enjoyed his performance after the former Celtic academy player received a reception from the Ibrox crowd. The Scotland captain had little to do as Salah scored three goals quickly.
Diogo Jota (Firmino, 73"), N/R -- Close to getting on the scoresheet but should have taken his effort first time. Harvey Elliott ensured that Liverpool would score their seventh.