First-half goals by Samir Nasri and Denilson settled the game and though the Belgians twice hit the woodwork they were second-best for most of the night.
Arsenal's victory takes them to 13 points, six clear of Olympiakos Piraeus after their goalless draw at AZ Alkmaar.
"In this group we were favourites and we just did the job," manager Arsene Wenger told a news conference.
"There are two competitions in the Champions League - one is a championship and one is a cup and now we move into the latter.
"That is much different psychologically and every minute becomes important."
Liege, with four points, need to win their last game at home to Alkmaar and hope Arsenal beat Olympiakos to allow them to progress on their better head-to-head record against the Greeks.
The Belgians survived some early scares on Tuesday, including one mad scramble in their goalmouth, but went close themselves after 21 minutes when Dieudonne Mbokani crashed a 20-metre effort against the bar.
Arsenal continued with their usual quick passing movements but the breakthrough goal was more Sunday league than Champions League as Liege defender Landry Mulemo slipped over while trying to deal with a hopeful punt by Thomas Vermaelen, letting Nasri through unopposed to clip in a neat half-volley.
Andrei Arshavin and Cesc Fabregas both shot inches wide and the match was effectively settled in first-half injury time when Denilson's 30-metre shot swerved beyond Bolat's hands.
"It was typically Brazilian, a floater, I think maybe they have shorter ankles or something that enables them to hit the ball higher on the laces," said Wenger.
Arsenal had come back from 2-0 down to win the teams' first meeting in September 3-2 but there was no sign of Liege, without key striker Milan Jovanovic, matching that, even though they had the clearer chances in the second half.
Axel Witsel missed one of them from close range while substitute striker Gohi Bi Cyriac hit a post late on.
In between, Arsenal played the ball around in neat patterns without really looking dangerous and the only thing for the crowd to get excited about was an 87th-minute red card for Mehdi Carcela-Gonzalez for a hint of a head butt on Fabregas.
"That red card will have serious consequences for us and I'm really angry as his lack of respect for his team mates and his club," said Liege coach Laszlo Boloni.
"He has to understand the responsibility of the club is more important than his own role - I've had a word with him."
Arsenal, who face Chelsea in a big Premier League match at the weekend, could be without William Gallas who hurt his ankle and got a black eye in a bloody collision with team mate Arshavin.
More seriously full-back Kieran Gibbs will be out for months after breaking a bone in his foot late in the game.
"He has big hole in his leg," Wenger said. "It doesn't look too good for him and it was a bad challenge."
