Klaebo's victory on Wednesday, which he clinched alongside teammate Einar Hedegart, ties him with US speedskater Eric Heiden's record five golds in a single Games. He had already surpassed the record for most titles at all Winter Olympics with his win in the men's relay.
France had been a podium contender going into the race, but suffered a series of setbacks after three-time silver medal winner Mathis Desloges broke his ski poles twice, landing the team in 12th place.
Norway held a commanding lead for most of the race, leaving Gus Schumacher from the United States and Italian skier Federico Pellegrino to battle it out for silver at the end. Schumacher finished 1.37 seconds behind Klaebo, who anchored Norway's team.
Italy's bronze is the second for Pellegrino, who also took third in the men's relay and said these will be his final Olympics.