The fan, identified by Ajax as a 19-year-old who had been drinking, approached Alkmaar's Esteban Alvarado from behind, but the Costa Rican keeper saw the supporter in time and skillfully tripped him.
Immediately afterward, Alvarado, enraged by the unexpected and unprovoked attack, kicked the supporter twice where he lay before security guards dragged the intruder off the pitch.
Referee Bas Nijhuis then gave Esteban a red card for violence. Alkmaar coach Gertjan Verbeek ordered his side to walk out in protest.
"I understand that Esteban was defending himself, but he walked to him (the supporter) and kicked him multiple times," Nijhuis said, defending the decision to send the goalkeeper off. "He could also have walked away."
Ajax was leading the round-of-16 match 1-0 after a goal by Gregory van der Wiel. Attempts to convince Alkmaar to continue were fruitless.
Alkmaar's players "didn't feel safe anymore in this situation," said club director Toon Gerbrands.
Ajax apologized for the security lapse and said the supporter will receive a lifetime ban from games.
Dutch football association director Bert van Oostveen said his organization is considering the incident, but is not certain whether the game will be replayed.
"Maybe I would have done the same thing, maybe not," said Ajax coach Frank de Boer, a renowned former defender for Netherlands' national team, of Alvarado's reaction. "But emotionally I do understand it."