The triumph means France matched a feat only previously achieved by the Soviet Union in 1982, by holding both the women's and men's world titles at the same time.
The sole survivor from the French women's previous world title in 2002 is coach Olivier Krumbholz, who was fired in 2013 but brought back two years later.
Norway, who have dominated women's handball in recent years, built a three-goal lead in the first half, but with goalie Amandine Leynaud excellent, France fought back to lead 12-10 at the break despite finishing the half playing five against seven.
In 'money time' at the end of a breathless match, 20-year-old Orlane Kanor and Allison Pineau made the difference with two goals each before Alexandra Lacrabere scored the final goal with 20 seconds to play.
"It's extraordinary! I'm very proud of us," Pineau said.
The victory was a case of third time lucky for France, who lost to Norway in World Cup finals in 1999 and 2011.
Norwegian captain Stine Bredal Oftedal did not hide her disappoint at the defeat.
“France hit us hard. They managed to kill our high tempo, and were also very physical in defence. On the occasions we broke through, we also missed a little too often. That's not what we wanted in a final,” Oftedal told NRK.
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