Norway secured the new record by winning bronze in the inaugural alpine skiing mixed team event as Switzerland took gold.
Norway passed the United States' tally of 37 medals, won at the 2010 Vancouver Games, with victory over France in the bronze-medal decider.
Switzerland beat Austria for gold in the event which sees four-strong teams compete in head-to-head slalom races.
Great Britain were knocked out in the quarter-finals by Norway.
Norway, led by their incredible cross-country skiers, have enjoyed a barnstorming Olympics. One day before the closing ceremony, they top the table with Germany on 13 golds, but with a vastly superior overall medals total.
Norway's haul was also boosted when Kristin Skaslien and Magnus Nedregotten were promoted to bronze in the mixed doubles curling, after Russia's Alexander Krushelnitsky failed a drugs test.
The International Olympic Committee's executive board met behind closed doors on Saturday to decide whether to lift the suspension of Russia despite two of its athletes failing drugs tests.
The latest offender, women's bobsledder Nadezhda Sergeyeva, has been kicked out of the Games after failing a test for a banned heart drug, the Court of Arbitration for Sport said.
In happier news, Finland won its first gold medal in Pyeongchang as Ivo Niskanen stormed across the line in the men's 50km cross-country skiing.