The Dutch international, who joined his Turkish employers from Liverpool earlier this summer, linked with strike partner Moussa Sow to level up proceedings at one goal apiece in the Russian capital with 65 minutes on the clock.
Kuyt's strike, which was his fifth for his new club, had cancelled out the opening goal of the match for the home side courtesy of a strike from Emmanuel Emenike after 59 minutes.
Travelling Fenerbahce were only level for four minutes though as Dmitriy Kombarov reacted quickest inside the area to put the hosts back in front.
Despite some late pressure from Fenerbahce, Kombarov's goal proved to be a winning one for Spartak.
The return leg will be played next Wednesday in Istanbul.