Salah, who by his own high standards has had a mixed start to the season, came off the substitutes bench in the 68th minute and seven minutes later started scoring.
First, he pounced on a loose ball and clinically slotted the ball into the net; then, he fired his second into the goal despite the attentions of three Rangers players, and suddenly he was on a hattrick.
And as quickly as Salah had scored his first two, he completed his third, curling the ball past Rangers goalkeeper Allan McGregor.
“Typical Mo [Salah],” Liverpool manager Jurgen Klopp said afterwards, according to Sky Sports.
The result, and Salah’s performance, provided some much needed relief for Liverpool.
A team that came close to winning the quadruple last season, has endured its worst start to a domestic season since 2012-13, so much so that Klopp declared his side’s title hopes already over at the weekend following a 3-2 defeat by Arsenal.
“It changed the mood definitely,” added Klopp of Wednesday’s win, which was the first time Liverpool had won an away game this season.
“It’s completely different. We usually drink a beer after away games but it’s that long ago I drunk a beer I’ll probably be drunk after one.”
Liverpool started poorly in this match and, as so often this season conceded the first goal, Scott Arfield putting the Scottish side ahead after 17 minutes.
A Roberto Firmino header from corner pulled Liverpool level, a though it needed a block from Liverpool’s Ibrahima Konaté to prevent Rangers from restoring their lead.
After halftime, Liverpool found some rhythm and Firmino extended the visitors’ lead with his second goal, latching onto Joe Gomez’s cross.
Ten minutes later, Darwin Núñez completed a counterattack after Firmino set up the Uruguayan international with a clever touch.
Then, after Salah’s six-minute hattrick, Harvey Elliott scored his first ever Champions League goal following a nervous VAR wait to complete the 7-1 victory.
Liverpool will next play in-form Manchester City in the Premier League at Anfield on Sunday.
“We all know who we are welcoming on Sunday,” Klopp said, per Sky Sports, “and this will be a different game but it’s better to go into it with this feeling we’ve got tonight than any other.”