The Germans, driven by Gotze who had a hand in each of his team's goals, looked capable of scoring every time they broke as they raced to a 3-0 half-time lead against the ineffective four-times former champions.
Marco Reus put the visitors ahead after eight minutes, combining with Gotze in a deadly counter-attack down the right before slipping the ball between goalkeeper Kenneth Vermeer's legs.
With Dortmund needing only a point to go through with a game to spare, Gotze added the second in the 36th minute and Robert Lewandowski scored from a rebound five minutes later after Gotze's initial shot was blocked.
Although Ajax had plenty of possession, Lewandowski added his second in the 67th minute with a tight angled shot after Gotze had found him unmarked.
Daniel Hoesen scored a consolation goal for the hosts with four minutes remaining but by then the damage had been done as Dortmund joined group runners-up Real Madrid in the knockout stage.
Dortmund coach Jurgen Klopp acknowledged Gotze's important part in proceedings but reckoned the 20-year-old could have done even better.
"Mario played exceptionally but also like we know he can play," Klopp told a news conference.
"But he could have done better in finishing."
Ajax, who might therefore be glad not to have suffered even more, never really troubled the Dortmund defence despite the possession they were allowed.
"We got a lesson in efficiency from Dortmund but I am also very angry about the way we defended," Ajax coach Frank de Boer told reporters.
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