Turkish Super Lig leaders Galatasaray made the short journey brimming with confidence having not lost their previous 12 league matches.
But a tight contest was settled in the closing stages as former Netherlands international Kuyt found the bottom left-hand corner with an angled finish.
Galatasaray's long wait for a win at their rivals subsequently continued with their last victory coming in 1999.
Hamza Hamzaoglu's men still lead the way, but Fenerbahce closed the gap to within just one point.
Second-placed Besiktas moved level with Galatasaray as Atiba Hutchinson's strike on the stroke of half-time was enough for a 1-0 victory at lowly Sivasspor.
Also on Sunday, strugglers Karabukspor were held 0-0 at home by Gaziantepspor, while it was the same scoreline in Istanbul Basaksehir's fixture against Konyaspor.
On Saturday, Guy Landel was on the scoresheet twice as Genclerbirligi ran out 4-2 victors at Kayseri Erciyesspor - who remained in the bottom three after finishing the match with 10 men thanks to John Boye's red card.
Rock-bottom Balikesirspor were denied a rare victory after throwing away a two-goal lead in a 2-2 draw with Rizespor, while Eskisehirspor and UEFA Europa League hopefuls Bursaspor played out a goalless draw.
Akhisar Belediyespor started the weekend by fighting back to draw 2-2 at Kasimpasa on Friday, while the action was rounded off on Monday by Trabzonspor's thumping 5-1 triumph at Mersin Idmanyurdu - Sefa Yilmaz scoring twice.
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