Ryan Donk put Arveladze's men ahead in the 19th minute, but their coach demanded his side immediately allow Konyaspor back in the game as Ryan Babel had been laid injured for the hosts in the build-up.
Hasan Kabze jogged through a static Kasimpasa defence to equalise from kick-off before scoring a penalty five minutes before the interval.
"Anyone would make same decision as me," he said. "I decided to make this choice because this is what Kasimpasa really are.
"I talked with my guys after match and told them that I'm going to resign. Thank you very much to everybody in club."
Slaven Bilic's men threw away a one-goal lead to lose 2-1 in Belgium on Thursday, but Gokhan Tore had Besiktas ahead in the 25th minute at Ataturk Olympic Stadium on Saturday with a remarkable looping shot from 30 yards.
Royston Drenthe smashed in a 30-yard free-kick to pull one back for the visitors, before Mustafa Pektemek bundled in Besiktas' fourth and Ramon's deflected strike made it 5-1 11 minutes from time.
In the weekend's remaining fixture, Gaziantepspor picked up a 2-0 victory over Trabzonspor.
