GANGNEUNG, SOUTH KOREA—For John Shuster, it wasn’t about the feel of the ribbon around his neck, or the surprising heft of the gold disc that lay on his chest. He had received an Olympic medal before, a bronze in 2006 as part of Pete Fenson’s team at the Turin Winter Games.

Shuster never forgot how it felt that night in Italy, standing on an Olympic podium and hearing a national anthem that was not his own. Saturday night, the curling skip from Chisholm, Minn., finally got his chance to sing. After a stunning, improbable run to America’s first Olympic gold medal in curling, Shuster put his hand over his heart, waited for the music to start at Gangneung Curling Centre and belted out the words that had waited inside him for 12 years.

He delivered the biggest shot of a 10-7 victory over Sweden, a brilliant double takeout that scored five points in the eighth end to break a 5-5 tie. Though it wrote a poetic ending to Shuster’s hard path to gold, he made it clear he was no soloist.

The U.S. won its second Olympic medal because of the harmony Shuster found with teammates Tyler George, John Landsteiner and Joe Polo of Duluth, and Matt Hamilton of McFarland, Wis. The best team he ever assembled shrugged off a 2-4 start to the Olympic round robin to win five in a row, ending its Winter Games with tears, hugs and a gold medal few saw coming.

“I was happy to get a chance to make that last (shot) for these guys, for all the shots they made through the course of this week and the course of this game,’’ said Shuster, 35. “I can’t tell you how un-nervous I was, sitting in the hack to throw it. These guys, their belief and their hard work, gave me the confidence to sit in the hack and let it go.

“I wanted to sing my national anthem and stand on the top of the podium at the Olympics. From the day the 2014 Olympics came to an end, every single day was with this journey in mind.’’

By the time he got to the end, it felt as if the whole world was with him. Ivanka Trump and presidential press secretary Sarah Huckabee Sanders showed up at Saturday’s game. So did King Carl XVI Gustaf, king of Sweden, and all the women of the U.S. hockey team that won a gold medal Thursday.

Thousands of people in the U.S. gathered to watch the game in the early-morning hours. Mr. T, who created the hashtag #curlingiscoolfool to express his new love for the sport, called the team Saturday morning to wish it luck; so did Olympic speedskating hero Dan Jansen, whose struggles inspired Shuster to work through his own.

They saw the U.S. take it to a Swedish team ranked No. 1 in the world. Sweden’s skip, Niklas Edin, had led his team to a 7-2 record and the top playoff seed. But even he sensed the Americans might finish off their run with the ultimate surprise.

“When they play well, they really get a good team spirit going,’’ said Edin, who routed the U.S. 10-4 in the round robin. “They had nothing to lose. We had everything to lose. The whole game, they played better than us.’’

Shuster came to the Pyeongchang Games as the first U.S. man to make four Olympic curling teams, but the previous teams he skipped never found their footing. The Americans were 10th at the 2010 Vancouver Olympics and ninth at the Sochi Winter Games in 2014. Shuster became a punching bag on social media, and when USA Curling held tryouts for its new national team program, he was cut.

He formed his own team, which has been the best in the U.S. for the past four years. It remained to be seen, though, whether Shuster could purge the ghosts of his Olympic past. It didn’t look promising when two bad losses dropped the U.S. to 2-4 in the round robin.

“After that, I sat on this grassy knoll out here, and I looked at the Olympic venue,’’ Shuster said. “And I was like, ‘This is silly.’ I’m getting my heart broken by this sport.

“I woke up in the morning and saw a story on Dan Jansen, who waited till his last moment to write his story. He had the same thing, and he got back up and changed his story. I’m so proud I was able to do something similar.’’

Jansen won his only Olympic medal in his third Winter Games. With no other alternative, Shuster’s team decided to just have fun and see where that took them. It never lost again.

The Americans scored two victories over Canada—which they had never beaten at the Olympics—on their way toward a gold-medal matchup with Edin. The more relaxed they got, the more their shots found their mark, a pattern that carried into Saturday’s game.

The U.S. and Sweden went back and forth in the first six ends, with neither team leading by more than a point. As the tension grew—even the team clown, Hamilton, had begun biting his nails—Edin began to miss some shots. Shuster had a couple of misfires as well, but when his teammates filled the house with American granite in the eighth end, he delivered.

“We didn’t get the rocks exactly where we wanted,’’ Edin said. “When I missed, we knew we were going to lose.’’

Edin left Shuster a window to knock out the two Swedish stones in the house, and Shuster’s perfect shot got them both, leaving five American stones in scoring position and blowing the game open.

After rolling with the punches all week, the Americans didn’t let a mix-up at the medal ceremony bother them. Shuster and his teammates were mistakenly given the gold medals engraved for the women’s competition.

All they really cared about was the color, and the song that would follow.

“For five days, we were the best team in the world,’’ George said. “We always knew we had it in us. To do it when it mattered most is what I’m most proud of.’