The Ghanaian side announced the arrival of the Ugandan international on a deal which runs until 2022, saying his addition would be vital for their Caf Champions League campaign.
Abege signed the contract to join Asante Kotoko after Sunday's friendly match between Sharks and Everton in Nairobi.
"George Abege's International Transfer Certificate (ITC) has reached the Porcupines, and the player is expected to join his colleagues this week," the club announced.
"The Porcupines noticed [George] Abege in the Confederation Cup preliminary round tie against Kariobangi Sharks, and find his qualities of relevance as the side aims for domestic and continental glories next season.
"Abege has great aerial abilities and ball control coupled with predatory skills that, on a good day, can make him every defender's nightmare on a good day."
Abege joined Kariobangi Sharks in June 2018 from Sony Sugar and scored three goals in the 2018/19 KPL season.
He found playtime hard to come by as Kariobangi Sharks' head coach William Muluya preferred to use Erick Kapaito and Sydney Lokale instead.