Les Bleus, back in the mix since Galthie took over last year, scored a try through Charles Ollivon on their first incursion and defended stoutly in a tense finale to snatch their second win in two games in this year's championship.
France, who host the 2023 World Cup, top the standings with nine points, ahead of Wales on goal difference with England in third place three points adrift.
"We were prepared to win the game, we won the game. It was our objective. That's it," Galthie, who has led France to nine victories in 11 games, told a news conference.