This was a notably more pleasing performance and result for the hosts after their 3-3 home draw with an admittedly excellent Brighton and Hove Albion in the Premier League on Saturday.
Alexander-Arnold and Salah’s conversions ensured their dominance was rewarded, and they would have won by a considerably greater margin had it not been for Allan McGregor’s brilliant goalkeeping display for the visitors – which saw the 40-year-old make eight saves.
Liverpool now have six points and sit second after three matches in the group, following their 4-1 opening defeat at Napoli and 2-1 Anfield win over Ajax.
While Rangers remain fourth having lost 4-0 in Amsterdam and 3-0 at home to Napoli.
The Italians are still top, having continued their 100% record with a brilliant 6-1 win at Ajax on Tuesday.
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Story of the match
Both sides made three changes from their respective domestic outings on Saturday – and switched shape.
For the Merseysiders, Fabinho, Roberto Firmino and Fábio Carvalho made way from the lineup that started the Brighton draw, with Darwin Núñez, Diogo Jota and Luis Díaz coming in. That saw Klopp opt for a formation more akin to 4-2-4 or 4-2-3-1 than the regularly used 4-3-3.
Following their 4-0 Premiership win at Heart of Midlothian, meanwhile, the Glasgow-based side brought out Ryan Jack, Scott Arfield and Antonio Čolak for Steven Davis, Malik Tillman and Alfredo Morelos. 3-4-3 was the approach Van Bronckhorst opted for.
With home and away supporters contributing to a crackling Anfield atmosphere, little time was wasted in matching that intensity on the pitch – particularly by those in red.
Within three minutes, a snapshot in the box from the lively Núñez forced the first of several sharp saves out of visiting goalkeeper McGregor.
Kostas Tsimikas sent a cross-shot behind shortly afterwards and then, in the seventh minute, Alexander-Arnold opened the scoring in sublime fashion.
Núñez did well to win a free-kick 25 yards out with his back to goal and the 23-year-old right-back made the most of it as he curled a brilliantly precise effort over the wall, beyond the reach of McGregor and into the top-left corner.
Even at this early stage, it was a goal that felt like it had been coming – and Liverpool kept coming.
The familiarly effervescent Díaz cut inside from the left and sent a shot just beyond the top-right corner five minutes later, shortly prior to Salah seeing a left-footed attempt tipped wide by McGregor after the Egyptian drove in from the opposite flank.
Former Liverpool man Ryan Kent fired well over from range at the other end in the 19th minute, with Rangers’ first attempt, but the tide continued to largely flow in the other direction.
Núñez took down a 34th-minute Jordan Henderson pass superbly in the box and had a sidefooted attempt palmed away well by the increasingly busy McGregor, who gathered a low Díaz effort moments after.
Two further impressive stops from the 40-year-old ‘keeper followed shortly before the break as he kept out Núñez attempts in the 41st and 43rd minute – after the Uruguayan made clever darts to latch on to well weighted Salah and Jota through-balls.
The summer signing curled a 25-yard effort narrowly over four minutes after the restart, but the lead was doubled three minutes later.
That was largely courtesy of Díaz, who controlled a nicely flighted Henderson pass, weaved into the box, darted between Leon King and John Lundstram and was brought down by the former.
Referee Clément Turpin swiftly pointed to the spot and, in front of the Kop, Salah coolly sent the penalty down the middle as McGregor dived to his left.
A two-goal advantage well earned, and it was only another excellent stop from the Scot – arguably his best of the lot – that prevented it becoming three in the 59th minute when he tipped a rasping Jota shot over.
Substitutions punctuated a quieter next 25 minutes, prior to Joël Matip heading just wide from a corner, and Rangers then had their most promising moments of the evening.
They chiefly came from substitutes, too.
First, Glen Kamara’s 85th-minute pass sent Rabbi Matondo into space down the left and the Wales international burst beyond Matip and into the box before his right-footed effort took a slight touch off goalkeeper Alisson Becker and was brilliantly cleared behind – and away from the waiting Fashion Sakala – by Kostas Tsimikas.
Then, from the resulting corner, Sakala’s header was blocked by Virgil van Dijk and Antonio Čolak’s close-range left-footed volley was superbly kept out by Alisson.
McGregor made his eighth and final save of the night from a curled Salah effort in the first minute of stoppage time, but at least there had been a couple of dangerous advances from the visitors to additionally wet the appetite for next Wednesday’s return fixture at Ibrox.
Rangers now know that they’ll need a win in that one to keep their hopes of reaching the knockout stages alive, while this was an Anfield outing that represented a considerable improvement on the weekend for the Reds.
A healthier way for them to head into Sunday's enticing looking Premier League trip to leaders Arsenal.
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Alexander-Arnold sets the tone
The No.66’s free-kick was a joy to watch. Pace, trajectory and direction were all wonderfully on point and it ensured that Liverpool’s fast start bore fruit.
This was only the third time that Klopp’s team have opened the scoring in a match this season and it laid the foundations for an impressively dominant night’s work.
A night on which Alexander-Arnold himself – and others – thrived off the team’s additionally strong foundations.
The 4-2-4 shape often saw Henderson drop into the space to the right of Matip when the hosts were in possession and that seemed to offer a welcome bit of extra protection and balance.
That helped the team sustain attacks more efficiently and gave Alexander-Arnold – the subject of considerable focus in recent weeks – a platform to make several of his typically quality-ridden offensive contributions.
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PLAYER OF THE MATCH: LUIS DÍAZ
Several of those in red were genuinely excellent within a highly effective collective display, and Díaz was a notable contributor.
The Colombian’s relentlessness from the left was crucial in ensuring Liverpool’s threat was a consistent and varied one.
Willing to go inside or outside, his thrilling directness was most clearly on show when he won the penalty. He faced up King and swiftly worked his way into a position which may well have led to him finding the net had the centre-back not brought him down.
Another eye-catching and effective display from the 25-year-old, who continues to represent a leading performer for the Merseysiders.
It would also feel amiss not to offer a very honourable mention to McGregor, who made so many superb saves.