After graduating from Jeonju University, Kwoun signed his first professional deal with Jeonbuk Hyundai Motors in 2006 and quickly displaced team-mate Lee Kwang-suk as first choice goalkeeper.

Kwoun made 43 first team appearances in his first season, most notably winning the 2006 AFC Champions League after beating Al-Karamah SC in the final.

Kwoun played a key role as Jeonbuk went on to win back-to-back league titles and would have won three in a row if they were not deducted nine points for their part in a scandal which involved bribing referees during the 2013 season.

He was however in goal for the season-opener, as Kwoun won his first trophy with Kashima in his first game after they lifted the 2017 Japanese Super Cup with a 3–2 win over Urawa Red Diamonds.

[20] Kwoun went on to make another four appearances for the national team during the qualification round of the 2018 World Cup, only conceding three goals in total.

His final international appearance was in a 3–2 defeat to Qatar in June 2017, which led to the sacking of then South Korea manager Uli Stielike.