Kwarasey played youth football with Trosterud and Vålerenga. After moving to Strømsgodset, he made his debut on 6 May 2007. He scored his first senior goal in 2010, converting a penalty kick in the cup against Modum. Kwarasey was the captain of the club in 2013 when Strømsgodset won Tippeligaen for the first time since 1970. He also won the Kniksen Award as the best goalkeeper in the 2013 season.