The home side began the game well and they took the lead in the 10th minute when Arkadiusz Milik's effort from distance struck the post, but bounced back to the feet of Anwar El Ghazi, who slotted the ball into the net.
Go Ahead pushed forward again and they should have taken the lead when Jeffrey Rijsdijk was allowed space and time inside the box to shoot at goal, but his effort went just wide of the target.
Ajax threatened just before the break when El Ghazi raided down the flank and into the box. The winger had the support of Milik, but decided to go alone and fired his effort over the bar.
In the second half, the home side regained their advantage when Jairo Riedewald was afforded time to lift a cross into the area for the run of Milik at the near post, who fired an effort past Erik Cummins.
Ajax secured their win in the 75th minute when Davy Klaassen won the ball deep in Go Ahead territory, and Lasse Schone latched onto the loose ball and drove into the box beating Cummins with an emphatic finish.