Poland defender Matty Cash was penalised for handball as the clock hit 90 minutes.
However, Depay blasted his penalty high and wide.
The Dutch went into Saturday's match in Rotterdam with a 100% record after a 4-1 win over Belgium and a late 2-1 victory against Wales.
Poland had beaten Wales 2-1 but suffered a 6-1 rout at the hands of Belgium.
After soaking up early pressure, the Poles opened the scoring after 18 minutes on Saturday.
Nicola Zalewski crossed from right where Aston Villa full-back Cash was on hand to slot the ball past Mark Flekken in the Dutch goal.
Five minutes into the second period, Poland were 2-0 ahead.
Krzysztof Piatek played a fine ball over the top to allow Przemyslaw Frankowski to break free of the Dutch offside trap.
He unselfishly squared the ball for Piotr Zielinski to tap in with the goal standing after a VAR review.
However, the Dutch roared back with two goals in three minutes with Davy Klaassen scoring from close range and Denzel Dumfries equalising from a Depay pass.
After Depay fluffed his penalty chance, he had the opportunity to quickly make amends but goalkeeper Lukasz Skorupski pulled off a sensational stoppage-time save.