Spurs fell 3-0 down inside 15 minutes at the Metropolitano stadium against Diego Simeone's side, with all three goals coming from mistakes.
Tottenham coach Igor Tudor selected Kinsky, a 22-year-old Czech goalkeeper who had not played since October, over Guglielmo Vicario after five straight Premier League defeats.
However, Kinsky slipped and gave the ball away in the sixth minute for Marcos Llorente to open the scoring.
After Micky van de Ven fell over allowing Antoine Griezmann to score the second, Kinsky erred again.
The goalkeeper bungled a pass and Julian Alvarez was able to walk the ball into the net.
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