Galstyan, 23, beat Japan's Hiroaki Hiraoka, silver medallist at last year's world championships, with an ippon, an automatic winning score, just 40 seconds into their final clash.
He joined jubilant Russian team mates in celebrations in front of standing ovation from a crowd which included a large number of disappointed Japanese fans.
Fourth seed Galstyan, 23, had earlier shocked Uzbekistan's Sobirov, who had been aiming to add an Olympic gold to his two successive world titles. Their close fight was only settled with a waza-ari throw in the extra time golden score period.
Sobirov, 25, a bronze medallist in Beijing four years ago, secured a second bronze by beating France's Sofiane Milous.
Brazil's Kitadai Felipe took the other bronze, beating Italy's Elio Verde.