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A first appearance in the competition's final for 25 years awaits Nice, with Nantes and Monaco going head to head on Thursday (AEDT) for the right to join it in Paris on 8 May.
Nice is owned now by British billionaire Jim Ratcliffe and has big ambitions it is moving towards fulfilling. A trophy is now in its sights, while it sits third in Ligue 1, which meant it began as overwhelming favourite against fourth-tier Versailles.
Goals in the second half from Amine Gouiri and Kasper Dolberg did the job, earning a 2-0 win for the home side as an initially nervy Nice stepped up the pace once it edged ahead.
"There was a lot of pressure before this match," Galtier said. "People wanted to see the amateurs beat the pros. The joy was not the same with us as it was after the successes against PSG or Marseille. It was more relief and satisfaction."
PSG and Marseille were beaten by Nice in the previous two rounds, and it will be the Parisians who are next to come to the Allianz Riviera on Sunday (AEDT) for a Ligue 1 assignment.
"We have to recover quickly to play a big match against Paris," Galtier said. "But there is a lot of joy and relief. I can't wait to see the Stade de France in red and black."
More joy for Galtier could come at the national stadium, as the man who won the Coupe de la Ligue with Saint-Etienne nine years ago and the Ligue 1 title with Lille last season attempts to complete the set of French football's major trophies.
"It would be great as a treble with three different clubs," Galtier said. "I've already won the Coupe de France as an assistant with Alain Perrin at Sochaux and Lyon, but as a head coach it would be extraordinary. However, we are a long way from that."
Goalkeeper Marcin Bulka said Nice was good value for its place in the trophy match.
"There are thousands of teams who take part in this competition, and we have qualified for the final!" he said. "We are very happy with this qualification. It's well deserved, because we eliminated strong teams like Paris and Marseille.
"It is no accident, no coincidence that we are here. The team deserves this victory and those before and to be in the final. Now we have one last step. We want to win and I hope we will achieve our goal."
