Ajax fullback Nicolas Tagliafico turned the ball into his own net in the 35th minute to ensure a winning start in Group D for the English champions, who looked the stronger of the two teams despite Ajax creating several good chances at the Johan Cruyff Arena.
Davy Klaassen struck the post with a powerful shot and Quincy Promes had a point-blank effort stopped by the legs of Liverpool goalkeeper Adrian as the Dutch club created a handful of excellent opportunities.
Fabinho moved from midfield to centre defence to replace Van Dijk and had an outstanding game, notably hooking a clearance off the line to deny Dusan Tadic just before halftime.
It was one of several chances for Ajax in the first half, as Lisandro Martinez had a header saved by Adrian and Ryan Gravenberch shot narrowly wide, much to the annoyance of Liverpool manager Juergen Klopp, who spent the early part of the game berating and shouting instructions to his players.
But his agitation was quelled when Sadio Mane turned the Ajax defence with a swift body movement to set up the goal. He miskicked his own effort, but it caught Tagliafico as he clumsily tried to clear, but instead deflected it into the Ajax net.
Ajax should have equalised, however, four minutes into stoppage time when Adrian did not properly deal with a cross into the area but substitute Jurgen Ekkelenkamp missed the target.