The Premier League champions were without the services of Virgil van Dijk, Alisson Becker and Thiago Alcantara for the clash.
The Dutch were force into an early change when Ghanian attacker Mohammed Kudus picked up a knock after just nine minutes.
Liverpool would go on to get the opening goal in the 35th minute when Nicolás Tagliafico deflected Sadio Mane's scuffed effort into his own net.
Ajax had a few sights at goal, with Davy Klaasen going the closest in the second half with a strike that rattled the upright.
Nevertheless, the Reds were able to see out the rest of the game to claim three points in Amsterdam.
Last season's quarter-finalists, Manchester City claimed a 3-1 victory over FC Porto in Group C.
While City were missing the likes of Kevin de Bruyne, Aymeric Laporte and Gabriel Jesus, Benfica took the lead on 14 minutes through Luis Diaz.
The Citizens were level five minutes later through Sergio Aguero's spot-kick after Pepe had brought down Raheem Sterling.
IIkay Gundogan completed the turnaround for City just after the hour mark before Ferran Torres sealed the win on 73 minutes.