Celtic centre-back Jack Hendry drilled in from 20 yards to give Scotland an early lead in Faro with his first international goal, but Depay soon equalised with a crisp volley.
Kevin Nisbet put the Scots back ahead after coming on as a substitute on the hour, also netting his first Scotland goal in just his second appearance.
David Turnbull curled just over and Craig Gordon made a fine save from Patrick van Aanholt’s deflected strike.
However the Scotland goalkeeper was rooted to the spot as Depay bent an 89th-minute free-kick around the wall to snatch an equaliser for the Dutch.
David Marshall, Stephen O’Donnell, Nathan Patterson, Grant Hanley, John McGinn and Che Adams all stayed behind at Scotland’s training base in Spain as a matter of precaution, with Fleck confined to his hotel room.
The Netherlands have also been affected by Covid-19, as goalkeeper Jasper Cillessen was dropped from the squad by coach Frank de Boer after testing positive last week.
Scotland will appear at the finals of a major tournament for the first time since the 1998 World Cup, while the Dutch return to the big stage after failing to qualify for Euro 2016 and the 2018 World Cup.
Steve Clarke’s Scotland face England, Croatia and the Czech Republic in Group D. The Netherlands are in Group C along with Austria, Ukraine and North Macedonia.
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