Van Persie replaced on-loan Liverpool winger Lazar Markovic on 77 minutes and scored the winning goal just two minutes later with his first shot on target.
The Netherland international has taken five shots in the league since joining Fenerbahçe and scored twice.
The 32-year-old striker now has three goals for the Yellow Canaries since joining on a three-year move from United over the summer.
Fellow former United teammate Luis Nani scored the Istanbul giants opening goal benefitting from a Markovic assist.
Nani has now scored four goals and provided two assists since joining the Istanbul giants.
FENERBAHÇE – BURSASPOR
Stadium: Ülker Stadyumu Fenerbahçe Şükrü Saracoğlu
Referee: Barış Şimşek
Kick-off: 18:00 GMT
Fenerbahçe expected starting XI: Volkan , Şener, Alves, Kadlec, Hasan, M.Topal, Ozan, Meireles, Markovic, Nani, Fernandao
Manager: Vitor Pereira
Bursaspor expected starting XI: Mert, Advincula, Serdar Aziz, Dany, Aziz , Faty, Hosogai, Dzsudzsak, Josue, Emre, Necid.
Manager: Ertuğrul Sağlam
Goals: 20′ Nani, 62′ Necid, 79′ Van Persie