Les Bleus triumphed on their own turf at Stade de France to throw down a gauntlet ahead of next year's home Rugby World Cup, where they will now be among the hot favourites.
Speaking after the full-time whistle confirmed they had edged Ireland for the crown, Galthie was ecstatic to get over the line two decades on from winning the championship as a player.
"After the frustration of second place, to be in first finally?" he said. "It's great. We're not doing somersaults yet because we're at a press conference, but they will come."
"There was a lot of tension during the match, because there was a lot of expectation. The symbolism of the Grand Slam was strong.
"We've not experienced this type of close match, [and] this tension jumped out at us. But it's an incredible first experience that will make us grow even more.
“It proves that we made the right choices, built the right team. It also means that we are on the right path with a team that is still young.
"The average age was 26 years old [on Sunday AEDT]. The team will continue to grow and progress until the [Rugby] World Cup.
"It's a long way between 2002 and 2022. But there is no coincidence. The path makes the destiny. Twenty years earlier, I am on the ground.
"Twenty years later, I am in the stands and I accompany the players to claim the trophy. It's nice to still be able to touch this feeling."
With the 2023 edition of the Rugby World Cup on home soil, France will be heading in pursuit of a maiden triumph, having finished as runner-up three times since 1987.