Liverpool’s head coach, Jurgen Klopp made six changes to his starting line-up, with Ibrahima Konate, making his first start of the season, among those to come in.
Andy Robertson was named on the bench following injury and returned to action as a second-half substitute.
A scrappy opening to the game was played out amid a deafening noise at Ibrox, but Elliott produced the first moment of real quality by spinning and driving goalwards from a central position and scooping a pass into the path of Nunez, who had strayed offside.
Rangers took the lead early on, when Arfield took advantage of the space afforded to him by placing an accurate shot from just outside the area into the bottom corner.
Seven minutes later, Firmino who has been on fire for Liverpool in recent matches equalized for the visiting side, after Kostas Tsimikas delivered an in-swinging corner and the No.9 glanced in a fine front-post header.
In the second half, Liverpool emerged after the break with renewed vigor and forged ahead 10 minutes into the half, when Joe Gomez’s inviting low cross was dispatched unerringly by Firmino for the Brazilian’s eighth goal of a productive campaign.
It was not long before Darwin Nunez made it three for the visitor, teaming up with Firmino to roll in a beautiful goal.
That was the Uruguayan’s final involvement as he was withdrawn for Salah, who was joined off the bench by Robertson and Thiago Alcantara.
Coming in, Mohammed Salah made history after driving in the fastest hat-trick in Champions League history in just six minutes and 12 seconds.
He first poked in cleverly from a tight angle, before then conjuring a couple of trademark finishes into either bottom corner to claim yet another goalscoring record.
Elliot made it seven for Liverpool, tucking home and finally able to celebrate his first Champions League goal following a lengthy VAR review that overturned an initial offside call.
