A minute silence was held prior to kick-off in tribute of the people that lost their lives during the tragic crowd crush which happened at a football match in Indonesia on Saturday.
It didn't take long for the home side to take the lead, Alexander-Arnold answered his recent critics as he emphatically scored Liverpool's first goal seven minutes in.
Alexander-Arnold has been subject to criticism after being a regular feature in the defence of Liverpool during their slow start to the season, but he struck a perfectly-placed 25-yard free-kick into the top left corner to give Klopp's side the lead against Rangers..
Liverpool made their quality known by forcing several chances in the first half, with Darwin Nunez coming particularly close.
And seven minutes into the second half, the hosts did double their lead when Mohamed Salah scored his fourth goal of the season.
Leon King, 18, hacked down Luis Diaz in the box to award Klopp's side a penalty and Salah converted with ease to provide the two-goal cushion.
Two chances in quick succession relied upon last ditch heroics from the hosts, Kostas Tsimikas cleared off the line before the resulting corner forced a good save from Alisson.
But Liverpool's victory wasn't troubled by Rangers, who have lost all three of their opening Champions League matches.