Georginio Wijnaldum gave Danny Blind's side the lead at the Philips Stadion, but goals either side of half time from Konstantinos Mitroglou and Giannis Gianniotas saw Greece come from behind to claim the victory.
It took the hosts just 14 minutes to open the scoring through Liverpool's summer signing Wijnaldum, although it was a fellow Premier League representative who created the chance as Tottenham Hotspur's Vincent Janssen showed clever footwork to reach the byline and fire a low cross in that Wijnaldum bundled home from close range.
A second nearly arrived shortly afterwards when Greece failed to clear their lines from a throw into the box, but Wesley Sneijder lashed his subsequent volley narrowly past the post.
The equaliser did arrive shortly before the half-hour mark courtesy of former Fulham striker Mitroglou, who planted a firm header beyond Zoet having been picked out by Vassilis Torosidis.
Netherlands were close to restoring their lead just five minutes later, though, and again it was Sneijder who saw a dipping shot flash just wide from range.
A raft of substitutions disrupted the flow of the game in the second half as both sides struggled to keep possession, and the first near miss of the period didn't arrive until the 67th minute when Janssen arrowed an effort inches past the post.
Greece somehow survived a goalmouth scramble shortly afterwards, but it was the visitors who took the lead with just over 15 minutes remaining when Gianniotas stroked home a rebound after Mitroglou had seen his effort saved by Zoet.
Janssen came close in the closing stages as Netherlands looked for another equaliser, but ultimately they fell short as Greece held on for a famous victory.
Netherlands will now begin their World Cup qualifying campaign away to Sweden, while Greece take on Gibraltar on Tuesday.