The Croatian striker, who this week won his appeal against a two-match ban imposed by UEFA for diving in a qualifying game, spoiled Liege's hopes of at least a point when he fired home from close range in the 81st minute from a Cesc Fabregas corner.
The Belgians had stunned the London side by grabbing two goals in the opening five minutes from Eliaquim Mangala and Milan Jovanovic.
A goal on the stroke of halftime from Nicklas Bendtner gave a lacklustre Arsenal hope for the second half. Defender Thomas Vermaelen levelled for the English side in the 78th minute before Eduardo grabbed the winner to spoil Liege's party.
Arsenal coach Arsene Wenger said he was shocked by his side's poor start but praised their second-half display.
"When you are a big club, you cannot accept such a defeat, so it was important to come back and win," he told reporters.
"We were caught cold by a Standard team that came out firing. It could happen to anyone, but to come back and win, showed a great team."
Liege coach Laszlo Boloni said Bendtner's goal just before the break was the turning point of the game.
"We were ready, we played very well in the first half, but were not very strong to continue in the second half," he said.
"It was a morale blow to take a late goal at the end of the first half."
TAKE CONTROL
Arsenal - who inflicted Liege's heaviest ever European defeat in 1993, triumphing 7-0 - found themselves behind after just two minutes when Mangala fired past goalkeeper Vito Mannone at the Champions League debutant's right-hand post.
William Gallas brought down Jovanovic just minutes later and the Serbian winger picked himself up to slot the penalty home.
Arsenal never looked like scoring in the opening half before Bendtner chased down the ball before rifling it between Liege goalkeeper Sinan Bolat's legs.
Arsenal began to take control after the interval and should have equalised earlier before Belgian international Vermaelen scrambled the ball past the goal line in the 78th minute following a Fabregas free kick.
Television replays suggested Arsenal were lucky to be awarded the goal as Alex Song appeared to handle the ball before setting up Vermaelen.
Another lapse in Liege's concentration allowed Eduardo to turn and fire home Fabregas' corner three minutes later and silence the home crowd.
