Liverpool outclass Rangers to quell early season crisis
Jurgen Klopp's men had won just three of their first nine Premier League and Champions League games of the season, but could not have asked for more obliging guests as an outclassed Rangers were happy to limit the damage.
Alexander-Arnold's stunning free-kick after only seven minutes could have opened the floodgates, but 40-year-old goalkeeper Allan McGregor kept Rangers in the game until half-time.
Salah's penalty early in the second period ended any doubt over the result to leave Rangers still waiting for their first point in Group A after three games.
Liverpool move three points clear of Ajax, who were thrashed 6-1 at home by group leaders Napoli.
All that was missing for Klopp was a first Anfield goal for Darwin Nunez as the Uruguayan missed a series of glorious chances on his return to the starting line-up.
Alexander-Arnold has also been under intense scrutiny for his lack of defensive qualities during Liverpool's slow start to the season.
But the right-back showcased his ball striking ability with a sumptuous free-kick into the top corner to settle any Liverpool nerves.
The concession of an early goal appeared to undo Rangers' boss Giovanni van Bronckhorst's game plan to defend in numbers in an attempt to frustrate the Reds.
Van Bronckhorst led the Glasgow giants to just their second European final in 50 years last season but they have struggled with the step up in their first experience of Champions League football in 12 years.
After conceding seven goals without reply in their opening two games, only the heroics of veteran McGregor prevented another rout.
Klopp handed Nunez just his second start since the Uruguayan was sent off on his home debut in August.
At a initial cost of 75 million euros ($73 million, £66 million), Nunez cost more than Rangers' entire revenue for the 2020/21 season, but the former Benfica striker's lack of confidence was obvious as he passed up a series of chances.
Four times McGregor thwarted Nunez before half-time as Liverpool went in at the break with just one goal to show for 45 minutes of complete dominance.
Any hope the visitors had of snatching their first point of the group disappeared shortly after the restart when Leon King was penalised was hauling down Luis Diaz inside the box.
McGregor had saved two Napoli penalties in a 3-0 defeat three weeks ago, but this time was outwitted by Salah as the Egyptian coolly slotted down the middle of the goal.
The chances continued to come for the home side, but they failed to fully cash in on what should have been a night to boost their confidence with a much more comprehensive scoreline.
Diogo Jota blazed over and Diaz prodded wide on the stretch after good work from Nunez down the right.
Liverpool can ill afford to be so profligate when they face leaders Arsenal and Manchester City in their next two Premier League games.
However, they are well on course to reach the last 16 of the Champions League for a sixth consecutive season with four points from their next two games away at Rangers and Ajax enough to seal qualification with a game to spare.
Follow us on Twitter @AzerNewsAz