The Real Sociedad forward snatched the only goal of the game in the 14th minute after Switzerland tried to play the ball out from the back but failed miserably, leaving Oyarzabal with a simple finish.
Both were solid during an unexciting clash at the Alfredo di Stefano stadium at Real Madrid's training ground, in which Spain did enough to win but were far from their best.
Goalkeeper Yann Sommer tried to take a short goal kick but misplaced his pass to Granit Xhaka, with Mikel Merino nipping in to steal the ball and feed Real Sociedad team mate Oyarzabal, who finished with ease.
Fati, who became Spain's youngest ever goalscorer in their previous Nations League outing against Ukraine in September, might have doubled the lead early in the second half but his strike was blocked.
Oyarzabal then hit the side netting from close range as Spain started to create more chances and the game became stretched, but neither side were able to find another goal.