Kosir had not won a parallel giant slalom race before overcoming Patrick Bussler by 0.42seconds in the final in Asahikawa, but having finished second twice and third once in the other three events of the 2014-15 season, the 30-year-old clinched World Cup gold.
Kosir defeated Sebastian Kislinger, Alexander Bergmann and Stanislav Detkov on his way to the final.
Detkov trumped Christoph Mick by 0.60secs to claim third.
Austria's Marion Kreiner won the women's overall title despite being eliminated at the quarter-final stage in Japan.
Kreiner's compatriot Dujmovits triumphed in Asahikawa, overcoming Switzerland's Julia Zogg, while Ester Ledecker, who finished second overall, defeated Claudia Riegler to finish third on the day.