Nicolas Tagliafico's first-half own goal was the difference between the sides in an end-to-end Group D game short on midfield control at Johan Cruyff Arena.
Starting life without injured talismanic centre-back Virgil van Dijk in the Dutch capital, the visitor could have done without the early mix-up between Adrian and Joe Gomez that suggested its defensive organisation is set to suffer.
Davy Klassen struck the post but that was as close as Ajax came to salvaging a point.
Hearts were in Liverpool mouths again soon after as Adrian's attempt to launch the ball forward was closed down, leading to a headed chance for Lisandro Martinez that needed saving.
Another well-worked Ajax move then ended in Quincy Promes being picked out at close range, but he somehow failed to beat Adrian when a goal looked certain.
But, as is so often the case, the visitor capitalised on that let-off ruthlessly, going straight up the other end to force a Tagliafico own goal from a horribly miscued Sadio Mane shot that followed a sublime turn in the box.
The host responded well to that setback, and would have been level at the break had Fabinho not acrobatically cleared Dusan Tadic's lobbed attempt off the line with Adrian stranded.
That late scare and a general lack of control in the middle throughout prompted Jurgen Klopp to make a half-time change, with youngster Curtis Jones making way for captain Jordan Henderson.
But the substitution looked to have made little difference when Ajax danced its way through the Liverpool midfield before teeing up Klassen to strike the post with a thunderous effort just over a minute after the restart.
With attempts to shore things up failing, Klopp instead opted to switch things up in attack, calling Diogo Jota, Takumi Minamino and Xherdan Shaqiri from the bench to replace his starting front three.
However, those changes only served to largely disrupt the flow of the game rather than add any thrust for the visitor.
Consequently, both sides struggled to create much during a scrappy finish to the game that ensured Liverpool took all three points.