Neither team was able to break the deadlock for the first half at Istanbul's Ulker stadium.
In the 45th minute, Kayserispor were down to 10 men as Ugur Demirok was shown a straight red card after a VAR (video assistant referee) check.
Ramazan Civelek netted the winning goal with an amazing long-range free-kick at the stoppage time while Kayserispor advanced to the next round.
Gaziantep FK beat Kasimpasa 2-1 and VavaCars Fatih Karagumruk defeated Ittifak Holding Konyaspor 5-4 to move to the quarterfinals in two other matches on Tuesday.
