Defending champions Germany took silver, climbing from fourth after cross country on Monday and helped by a clear show jumping round from 2012 individual winner Michael Jung.
Australia took third while New Zealand, which was in the running for silver after cross country, fell to fourth after a seven-time Olympian Mark Todd knocked three rails down.
Eventing star Phillip Dutton delighted with his first Olympic medal in American colours
In addition to determining the team medals, the morning jumping round also served as a qualifier for a final jumping round to determine individual medals on Tuesday afternoon.
Cross country is a three-phased sport including dressage, cross country and show jumping. France last won team gold in the sport in Athens in 2004.
Elsewhere in the equestrian events Germany's Michael Jung claimed his second consecutive individual eventing gold medal on Sam after posting two clear rounds in the final show jumping phase on Tuesday.
Jung, widely considered the best eventing rider ever, only turned to Sam, the 16-year-old German Sport Horse he rode to win gold in London, a week before the animals were due to fly to Rio. His first and younger choice, Takinou, had suffered an infection.
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