Ersun Yanal's side's unbeaten run in the Turkish top flight looked set to end at 11 games at Sukru Saracoglu when a Hugo Almeida double put their city rivals 3-2 ahead at the break following earlier goals from Olcay Sahan, Emmanuel Emenike and Moussa Sow.
However, Netherlands international forward Kuyt had other ideas, as he levelled with seven minutes to go to ensure Fenerbahce took a share of the spoils, with the result leaving Besiktas in third.
Second-placed Kasimpasa missed the opportunity to close the gap on Yanal's pacesetters to four points when they were held to a 1-1 draw by champions Galatasaray on Sunday.
Shota Arveladze's men were on course for a fourth straight Super Lig home win when Egyptian striker Sanharib Malki put them in front after 11 minutes, only for Roberto Mancini's side to snatch a point - which lifts them to fourth in the table - through Burak Yilmaz on the hour mark.
Meanwhile, Trabzonspor moved up to fifth with a 3-1 home win over struggling Kayseri Erciyesspor.
Former Chelsea winger Florent Malouda put the hosts in firm control with a brace, before Yasin Oztekin pulled one back for Fuat Capa's team on 59 minutes. Olcan Adin then eased Trabzonspor nerves by restoring their two-goal cushion nine minutes from time.
Sivasspor fell to sixth after slumping to a second consecutive league defeat when mid-table Antalyaspor pulled off a surprise 3-0 win at the 4. Eylul Stadium on Friday, while Eskisehirspor remain seventh after their 0-0 stalemate with Karabukspor on Sunday.
At the other end of the table, bottom-placed Elazigspor were beaten for the ninth match in a row as Bursaspor won 2-1 at the Elazig Ataturk Stadium on Friday courtesy of Taye Taiwo's early penalty and Fernando Belluschi's winner 20 minutes from time.
Second-bottom Kayserispor picked up a point by drawing 0-0 with Akhisar Belediyespor on Saturday, and Gaziantepspor ran out 3-1 winners at Genclerbirligi - thanks in no small part to a double from Abdou Razak Traore.
In Monday's only Super Lig fixture, Konyaspor overcame Rizespor 2-1 at home after Godfrey Oboabona went from hero to zero for the visitors.
The Nigeria international looked to have snatched a point for Rizespor when he equalised Omer Sahiner's opener in the 73rd minute, but then gifted the hosts the win by putting through his own net with nine minutes remaining.
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