Home captain Memphis Depay missed a late penalty that would have secured all three points for his side as he struck a post, but the result sees the Dutch top Group Four with seven points from their three games, three ahead of Belgium and Poland.
The Poles, seeking a first ever victory against their hosts in the Netherlands, led 2-0 just after halftime as English-born Matty Cash and Piotr Zielinski scored, much to the delight of their sizeable contingent of travelling fans in the stadium.
But a frenzied five minutes in which Poland got their second goal and the Netherlands responded quickly with a double strike turned the game on its head, before Depay wasted the late chance to complete the home side's comeback.
Netherlands host Wales on Tuesday in their final game of this international window, while Poland welcome Belgium on the same evening.
"I missed and unfortunately that can happen," Depay told NOS. "Now it feels like a defeat. It was a difficult game. Poland were compact and the spaces were super small.
"But we have shown that we have fight in us. We continue to believe in ourselves, but today was difficult.
"The 0-2 was a mental blow, but we handled it well. It wasn't our best game, but we kept fighting."
Poland were without talisman Robert Lewandowski as the striker sat out through injury, but took the lead on 19 minutes when Cash scored his first international goal.
He picked up the ball outside the box and fired low into the opposite bottom corner.
The visitors doubled their advantage just a few minutes into the second period when Przemyslaw Frankowski beat the offside trap and laid on a tap-in for Zielinski.
The Netherlands came up with an instant response though, Klaassen finishing from inside the six-yard box. And Dumfries made it three goals in five frenetic minutes as he blasted the ball into the net after Depay's pass.
The home side had an excellent chance to win it when Cash was adjudged to have handled the ball in the box.
But Depay, who was mostly poor all night, fired his penalty against the outside of the post, the first player ever to miss three spot-kicks for the Dutch