A defensive error cost the hosts an own goal early on, as flamboyant winger Armindo Bruma's pass into the six-yard box was sent into the net by Bursaspor defender Ertugrul Ersoy.
As the half-time whistle loomed, the Istanbulites' high-tempo play proved too much for Bursaspor, with Bruma finishing in the 45th minute for a 2-0 lead.
Galatasaray continued the scoring spree in the second half, with Ahmet Calik contributing in 58' and Lukas Podolski in 60', before Yasin Oztekin's goal decided the final score in 89'.
The win raises Galatasaray's points to 52, cementing their hold on the fourth place.
Bursaspor are stuck at the 13th spot with 35 points, leading manager Mutlu Topcu to announce his resignation shortly after the game.
Basaksehir beat Besiktas 3-1
In a battle atop the league on Saturday, second-place Medipol Basaksehir defeated leaders Besiktas 3-1 at home.
Nineteen-year-old Cengiz Under was in top form for Basaksehir as he scored two goals for the home team, which racked up three unanswered goals in the first 20 minutes with Emmanuel Adebayor chipping in the third.
Striker Demba Ba scored for Besiktas in 87' but it was too little too late for the Istanbul club.
The victory reduces the gap between Basaksehir and Besiktas to just four points, with five matches to go in the season.
Here are the full results of match week 29:
Gaziantepspor - Kardemir Karabukspor: 0-0
Kayserispor - Aytemiz Alanyaspor: 3-0
Trabzonspor - Genclerbirligi: 0-0
Kasimpasa - Akhisar Belediyespor: 0-2
Fenerbahce - Caykur Rizespor: 2-1
Adanaspor - Antalyaspor: 2-5
Medipol Basaksehir - Besiktas: 3-1
Bursaspor - Galatasaray: 0-5
Osmanlispor - Atiker Konyaspor: 0-0