The victory was the first for France in the tournament after falling 4-1 to England and drawing 1-1 with the U.S women. But it was costly, with defender Aissatou Tounkara stretchered off at the end of the first half with what appeared to be a serious leg injury.
Henry scored in the 10th minute. Le Sommer scored her 66th international goal in the 55th minute to give France a 2-0 lead before Gauvin's goal in the 68th.
France, ranked No. 6 in the world, started veteran Sarah Bouhaddi for her 129th appearance in goal for the national team.
The French are playing under coach Corinne Diacre, a former national team defender, as they prepare for the 2019 World Cup on home soil.
Germany fell 1-0 to the United States in the opening match of the She Believes Cup on Thursday in Columbus, Ohio, then played to a 2-2 draw with England in the second match at Red Bull Arena.
Ranked No. 2 in the world, Germany won gold at the 2016 Olympics. Afterward, longtime coach Silvia Neid stepped down and was replaced by former national team defender Steffi Jones.