SAITAMA, Japan — Gregg Popovich, the head coach of the United States men’s basketball team, exhaled on Saturday afternoon as he contemplated his team’s 87-82 victory over France to win an Olympic gold medal.

“Do you know what sayonara means,” Popovich said, smiling. “That’s how I’m feeling now.”

It was that kind of summer for Popovich and the American players, who are always expected to dominate international basketball, yet always explaining how hard it is to do so.

This time around, though, the difficulty felt palpable. The players described their road to the title as an odyssey. They were away from their families, far from home.

They had only a short time to get to know one another, just a few weeks to reach their top gear and fulfill their potential against a rapidly improving field of experienced international teams. They delivered in the end — the gold medal was the team’s fourth consecutive and 16th overall — but the result did not always feel assured.

The team lost two warm-up games in Las Vegas before flying to Japan for the Games. They dropped their opening game in Tokyo against France, raising questions anew about their level of preparation for these Games and their overall level of commitment to international basketball, which has some clear differences from the N.B.A. game, including shorter quarters and a 3-point line that is closer to the basket.

What came next was a gradual blossoming they hoped, but could not be sure, would come.

“When you’re part of a team that’s evolving by the second, it’s just amazing to see,” said Kevin Durant, who was the focal point and driving force of the team’s offense throughout the tournament. “I’m bonded with these guys for life. It’s family for life. I’m grateful we all committed to this early, stuck with it, and finished it off.”

Durant was spectacular again for the American team, scoring 29 points to go with six rebounds. Damian Lillard said it felt hard sometimes not to stand around and watch him play.

The French had held Durant to 10 points in their win over the United States in the group stage. But there is only so much a defense can do when Durant has attained a state of flow.

“We tried to make things tough for him, tried to make him work as hard as he can, but he’s Kevin Durant, and he’s going to hit some shots only he can hit,” said Rudy Gobert, the French center. “I think he’s the best scorer in basketball. He’s going to do what he does, especially on the biggest stage like the Olympic Games.”

The differences from the first game between the two teams were quite clear.

After that loss, Jrue Holiday admitted he was still getting accustomed to the ball used in international play, which has a slightly different feel from those used in the N.B.A. France moved like a well-oiled machine, whirling around an American defense that looked a half-step slow.

But on Saturday, whenever France made a move, the Americans were there to evade and counterpunch.

Vincent Collet, the longtime head coach of France, noted the evolution of the United States’ play between the two contests. In both meetings, he and his players tried to feed the ball into the paint, where they enjoyed a considerable advantage in size. But during the final, the American guards showed far more tenacity and pressure on defense, making each entry pass a mini adventure.

Lost in the swirl of the Americans’ athletic defense, the French committed 18 turnovers, compared to the United States’ 9.

“If we turned the ball over it was usually a bucket the other way — they feed off that,” said Gobert, who led France with 16 points and eight rebounds before fouling out in the closing seconds of the game. “With turnovers, missed three throws, those little details, you shoot yourself in the foot.”

If Durant was the sparkplug on offense, Holiday was the most menacing presence on defense.

After the game, Draymond Green likened each opposing team’s offense to a fence unable to stop Holiday.

“He bent every team’s fence from the time he stepped off the plane,” Green said, calling Holiday the best on-the-ball defender in the N.B.A. “He was extremely special, and we couldn’t have done this without him.”

Holiday and Khris Middleton won an N.B.A. championship with the Milwaukee Bucks on July 21. Seventeen days later they became gold medalists.

“That’s a hell of a summer,” Holiday said.

After a transcontinental flight, Holiday, Middleton and Devin Booker of the Phoenix Suns had arrived at the team hotel the day of their first game of the Olympics, embodying in a way the team’s harried preparations for these Games.

It was clear, then, that the Americans on Saturday afternoon were feeling a hefty amount of pride about the team’s progression from that opening night loss.

Durant said the players held a team meeting after that first game, without coaches, to reaffirm their commitment to getting better, to gelling as a group.

Lillard acknowledged there was a feeling of relief mixed with joy as the players celebrated on the court after the game, draping themselves in American flags.

“It’s an underappreciated thing,” Lillard said about the difficulty of throwing N.B.A. superstars together on a single national team. “It sounds good, but we’re competing against high-level players, and when they’re connected as they are, that makes it even tougher.”

“Finally getting there, pulling the gold medal game out, it’s like you almost kind of exhale,” Lillard said.

It was no surprise, then, that Popovich looked spent, ready for rest and refreshment, as he pondered the journey he and his players had completed.

“I’m looking forward to getting back to the hotel and having … something,” Popovich said, making Green and Durant laugh.