US forward Tyler Boyd scored the opener in the sixth minute to give Besiktas a 1-0 lead at Vodafone Park.
But Guinean forward Bengali-Fode Koita shined on the pitch, scoring two goals for Kasimpasa and making it 2-1.
The Black Eagles equalized in the 49th minute as Kevin-Prince Boateng put the ball into Kasimpasa's net.
Yalcin's header goal in the 90th minute brought victory for Besiktas.
With 53 points, Besiktas jumped to fourth spot in the standings of the Turkish Super Lig. Kasimpasa, with 36 points, is in 11th place.
League leaders Medipol Basaksehir have 66 points, while second-place Trabzonspor have 62 points.
Demir Grup Sivasspor, with 54 points, is in third place and Galatasaray, with 52 points, is in fifth spot.
Super Lig Week 31 results are as follows:
Tuesday:
Genclerbirligi - Fenerbahce: 1-1
Medipol Basaksehir - Yukatel Denizlispor: 2-0
Wednesday:
DG Sivasspor - BTC Turk Yeni Malatyaspor: 0-1
Trabzonspor - Antalyaspor: 2-2
Aytemiz Alanyaspor - Galatasaray: 4-1
Goztepe - MKE Ankaragucu: 2-2
Thursday:
Caykur Rizespor - HK Kayserispor: 3-2
Besiktas - Kasimpasa: 3-2
Gaziantep FK - IH Konyaspor: 3-1
