With Senegal missing injured star man Sadio Mane and the Dutch without their main attacking threat Memphis Depay at kick-off, the Group A encounter in Doha looked set to end in a goalless draw.
But with just six minutes remaining PSV Eindhoven forward Gakpo got up in front of Senegal goalkeeper Edouard Mendy to head in a floated Frenkie de Jong cross from the left.
Substitute Klaassen then secured the victory deep in injury time, following up to score after Mendy had saved from Depay, who had also come off the bench.
The result leaves the Netherlands level at the top of the group with Ecuador, who beat hosts Qatar 2-0 in the tournament's opening match on Sunday.
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