The striker scored three goals in his first 13 minutes on the pitch, quieting a once-euphoric Ibrox crowd.
Rangers took an early lead thanks to Scott Arfield’s goal, but a Roberto Firmino double and a Darwin Nunez score put Liverpool ahead, and Harvey Elliott added a seventh following a VAR review.
The Premier League team produced a mesmerizing second-half effort, and they now just need a draw against Ajax in Amsterdam in two weeks to go to the next round.
Rangers are still looking for their first points, and their ambitions now pivot to finishing third in Group A and qualifying for the Europa League. It doesn’t get any simpler for them, though.
It does not get any easier for them, though, as they travel to face group leaders Napoli next.
As Salah trotted on to the Ibrox pitch, it was with little fanfare. Nunez had just given the visitors a commanding 3-1 lead, and the Egyptian’s introduction seemed almost like the cherry on the cake after a tasty game had been yanked away from Rangers.
But a player whose stellar form has dipped to merely good this term produced something special.
It started with an old-fashioned toe poke from a tight angle. Salah was off and running. A handful of minutes later, a drop of the shoulder, a quick change of pace on the edge of the box, and a deceptive finish brought his second. Ibrox, not for the first or last time in the half, fell silent.
There was barely time to type his name as the player of the match before his astonishing intervention was complete with yet another composed finish.
After just 17 minutes of this frenzied contest, it looked all so different. All of Liverpool’s wins since August have come in Europe, but another looked unlikely amid a cacophony of noise in Govan.
A flowing Rangers move cleaved open a static Liverpool defence and allowed Arfield space to steer a shot low across Alisson and into the net. Finally Rangers had arrived in this year’s Champions League.
It was an arrival which was celebrated for seven minutes. Kostas Tsimikas’ whipped corner to the near post found an unmarked and grateful Firmino, who nodded in from six yards.
Ten minutes after the break, Firmino hushed a raucous crowd again. Joe Gomez’s rampaging run down the right created the opening, and his inch-perfect delivery found the Brazilian on the run for a low first-time steer through Allan McGregor’s legs.
The writing was on the wall for Rangers by this point, and it was absolutely game over on 66 minutes. Again Firmino was involved, his cute flick teeing up Nunez for a curling shot around McGregor from the edge of the box for a goal met with a ferocious punch of the air by manager Jurgen Klopp.
After Salah was finished stealing the show, there was still time for more as Elliott got in on the act. The youngster was originally flagged for offside after volleying in, before a VAR check reversed the decision and sent him jogging away in celebration to a jubilant band of supporters in the far corner.
Liverpool rally to send a statement – analysis
Liverpool needed this result, but the scoreline sends a message that they are still capable of being dominant.
Apart from Firmino’s opening, they were ineffective in front of goal in the first half. On the other end, they were also getting themselves into trouble.
However, the Brazilian was pivotal in changing the tide. His link-up play was entertaining to watch, and his finishing and impact were crucial as Liverpool ran rampant after the break. Then there’s Salah…
Rangers, on the other hand, are currently bottom of Group A with a goal differential of minus 15 and no points. As a winter of discontent builds on the continent, the fire of Seville and the Europa League final has long ago dissipated.
There is still time for Giovanni van Bronckhorst’s squad to achieve some type of European football after Christmas, but there won’t be many inside Ibrox who believe they will.