Salah, who was enjoying a mixed start to the season with his high standards, came off the bench in the 68th minute and seven minutes later began scoring.
First, he hopped on a loose ball and slashed it clinically into the net; Then he hit his second goal in the net despite the interest of three Rangers players, and suddenly he was on a hat-trick.
Once Salah scored his first two goals, he completed his third, bypassing Rangers goalkeeper Alan McGregor.
“Typical mo [Salah]Liverpool boss Jurgen Klopp said afterwards, according to Sky Sports.
The result and Salah’s performance provided some much-needed relief for Liverpool.
The team, which came close to winning a quadruple last season, faced their worst start to a domestic season since 2012-13, so much so that Klopp announced his team’s title hopes were already over at the weekend after the 3-2 loss to Arsenal.
“It definitely changed the mood,” Klopp added of Wednesday’s win, the first time Liverpool have won an away game this season.
“It is completely different. We usually drink beer after away games, but for a long time I have been drinking beer that I will probably drink after one.”
Liverpool started poorly in this match and, as so often this season, conceded the first goal, Scott Arfield put the Scottish side ahead after 17 minutes.
Roberto Firmino’s corner kick lifted Liverpool’s level, although it needed a Liverpool save from Ibrahima Konate to stop Rangers regaining their lead.
After the end of the first half, Liverpool found some rhythm and Firmino extended the visiting team’s lead with his second goal, rebounding in Joe Gomez’s cross.
Ten minutes later, Darwin Nunez completed a counter-attack after creating the Uruguay international Firmino with a smart touch.
Then, after Salah scored a hat-trick for six minutes, Harvey Elliott scored his first Champions League goal after waiting for the VAR to complete the 7-1 win.
The next Liverpool will play Premier League-style Manchester City at Anfield on Sunday.
Speaking on Sky Sports, Klopp said: “We all know who we’re going to welcome on Sunday, and it’s going to be a different game, but it’s better to go into it with that feeling we have tonight than any other game.”