Laura Kalmari gave the Finns an early lead in the sixth minute, taking advantage of a defensive error before slotting home.
Heidi Kackur's powerful header sent the 500 traveling Finnish fans into delirium just 10 minutes later as they looked to book a place in the last four.
Denmark pulled one back through Cathrine Paaske Sørensen but the Finnish defence were resilient and refused to cave in despite wave after wave of Danish pressure as they booked a Wednesday semi-final against the Group B winners in Preston.