"I liked the game a lot. We got three points. That's the important thing," Klopp said, looking happy but more subdued than usual after a European win that left Liverpool second in Group A on six points after three games.
"I can't wait for the moment when I can read newspapers again. But the criticism is completely fine. We are not over the moon about our situation.
"I saw tonight a team fully committed, that's what I like a lot. It was exactly what we needed tonight. A new formation, we had to change things. We had plenty of chances, imagine if we had taken more of them. Really good game. Very important."
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 midfielder Jordan Henderson said: "It felt good out there. You try to not listen to social media, especially when you're going through a tough period as a team. But it can be difficult, it can hurt.
"You have to try and find a way to use that as fuel when you get out on the pitch."
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.
"We are competing against one of the best sides in Europe," Rangers manager Giovanni van Bronckhorst said.
"The level we face is very high and the level we have today wasn't good enough to get something out of the game."