It was a deserved 24-23 success, as they held the lead from the 10th minute and will now definitely fight for a medal - meeting reigning gold medallists Norway in Thursday's semi-final.
For Russia, it marks the end of a turbulent competition where the squad seemed at odds with extravagant coach Evgeny Trefilov, who shouted orders from the touchline.
Indeed, the Russia stumbled through Group A, finishing third, but were still favourites to beat Republic of Korea, who had been in fine form during their preliminary stage.
It was the Koreans who started off the brightest, surging into a 6-2 lead midway through the first half, and that became 11-5 10 minutes later as they threatened to run away with it.
Russia had to come up with something and they did as four unanswered goals helped rescue a desperate situation and they went in at half-time 14-11 down.
They took that momentum into the second half and soon levelled thanks to three successive goals, but Republic of Korea would not lie down and managed to re-establish a three-goal lead at 21-18 with 11 minutes remaining.
But the topsy-turvy nature of the game continued and Russia enjoyed a dominant spell to again equalise at 22-22, before Korea again took the ascendancy with decisive back-to-back goals gave them a two-goal advantage.
The Europeans threw one in with 30 seconds remaining and were unable to make use of a free throw right at the death as the Koreans held on.