A second-half goal from teenager Curtis Jones ensured the Reds would progress to the next stage on Wednesday, although academy prospect Caoimhin Kelleher, given his chance in goal, was required to make a stunning late save from Klaas Jan Huntelaar to protect the points at Anfield.
Better was still to come for Jurgen Klopp and co, too, as news of Atalanta's shock draw with Midtjylland meant top spot was clinched in Group D with a game to spare.
Liverpool had been hit by yet another injury blow in the form of a hamstring issue for Alisson, meaning Kelleher was handed his Champions League debut, preferred to Adrian.
The 22-year-old was initially a virtual spectator during a strong start from the hosts that saw fellow youngster Jones crack the post from distance.
Ajax grew into the game as it progressed, however, and called Kelleher into action on the half-hour mark through Noussair Mazraoui's thunderous effort from the edge of the box.