"This is the first and now hopefully we can take more prizes," said defender Daley Blind after scoring the opening goal in Sunday's 4-0 win over Willem II Tilburg in the final at Rotterdam.
Ajax could still achieve a triple this season.
They take on Tottenham Hotspur on Wednesday for a place in the Champions League final and are top of the table in the Dutch championship with two rounds to play.
"It looks as if everything is going right for us, both on and off the field. It's a feeling we must treasure," Blind told reporters. "But we are not finished. The season can still get whole lot better."
Sunday's Dutch cup win was the 19th in total for Ajax but first since 2010. They were last Dutch champions in 2014 and have not won a European club prize since their Champions League success in 1995.
Ajax have a 1-0 advantage over Tottenham after last week's semi-final first leg in London as they seek a berth in this year's final in Madrid on June 1.
"The season has gone perfectly but we are not yet satisfied," said coach Erik ten Hag. "We have shown some good things up to now and those memories will always be there. Beating Real Madrid away... that we'll always remember.
"But realistically you need to have something tangible. And that is titles. Now that we have the cup, it must lead to others."
(Reporting by Mark Gleeson; Editing by Sudipto Ganguly)
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