The Black Eagles won their first home game with new manager Valerien Ismael at Vodafone Arena.
Michy Batshuayi led his team with two goals in the second and 43rd minute while the other goals were scored by Josef de Souza in the 25th minute and Guven Yalcin in minute 78.
Efecan Karaca scored the only goal of Alanyaspor in the 23rd minute.
Besiktas is sixth with 50 points in the standings while Alanyaspor is in the fourth spot with 52 points.
Leaders Trabzonspor loses two points against Gaziantep FK
The Black Sea Storms seem to have a drop in their Super Lig form as they have managed just one win, three draws and one loss in their last five Turkish Super Lig matches.
Second place Konyaspor also missed a chance to close the gap as they lost to Caykur Rizespor 1-2.
Trabzonspor still has the lead with 72 points -- a 14 point lead against Konyaspor, which has 58 points, while Gaziantep FK is 14th with 41 points in the Super Lig standings.
Turkish Super Lig Results:
VavaCars Fatih Karagumruk-Kasimpasa: 3-2
Goztepe-Yukatel Kayserispor: 1-2
Caykur Rizespor-Ittifak Holding Konyaspor: 2-1
Gaziantep FK-Trabzonspor: 0-0
Besiktas-Aytemiz Alanyaspor: 4-1
