The Premier League champions were toothless in the first half but finally clicked 10 minutes after the break when the impressive Trent Alexander-Arnold set up Diogo Jota for a tap-in.
A late penalty lashed home by substitute Mohamed Salah gave the scoreline unwarranted gloss as Liverpool went to the top of Group D.
“It was a tough night,” Liverpool manager Jurgen Klopp told BT Sport. “It’s a bit like in a marriage, there are good and bad times. It’s not a bad time but it’s a tricky time. A hundred percent we have to stick together and fight harder and that’s what the boys did tonight.”
The German boss said Fabinho’s injury was “exactly the last thing we needed”.
“He felt his hamstring so that’s not good,” said Klopp. “He didn’t feel it that much — he said he could’ve played on — but not sprinting so that doesn’t help. We will see, we will have to do a scan and see but clearly it’s not good.”