Ri, 21, scored 15.691 points to add the Olympic to his two world champion titles on the apparatus.
Russia's Denis Abliazin scored 15.516 to take silver for his second consecutive Games medal on vault after London 2012.
Japan's Kenzo Shirai snatched bronze ahead of Romanian veteran Marian Dragulescu.
Both scored 15.499 points with Shirai achieving a higher execution score.