Swarzak retired Javier Baez on a flyout to right fielder Avisail Garcia to strand two runners after entering in the eighth. Kris Bryant reached on an infield hit and Anthony Rizzo walked with two outs against him in the ninth, but Swarzak struck out Willson Contreras looking.
That third-strike baseball went to the veteran, who knew it was his first save.
"Absolutely. I've been waiting for that opportunity for a long time," Swarzak said. "It's nice that I went in there and got it done.
"You think about that moment for years, and then it finally happens. You just are trying to take a step back and reflect on what just happened, and I'll be able to come in tomorrow and be ready to go."
Newcomer Tyler Clippard was anointed by manager Rick Renteria as the closer when White Sox relievers David Robertson and Tommy Kahnle were traded to the Yankees. Swarzak, meanwhile, was being used in a true "fireman" role, as described by Renteria, meaning he was called on for the toughest spots at any point of the game.
Renteria called on Swarzak in the fifth inning Sunday at Kauffman Stadium to protect a one-run lead with two on and two out. The right-hander struck out Jorge Bonifacio on three pitches and went on to work 1 1/3 scoreless innings.
"That's where the game is going right now," said Swarzak, who has a 2.23 ERA over 41 games, with 52 strikeouts and 13 walks in 48 1/3 innings. "Relievers in general, just our stocks are rising as the years go on.
"It's kind of progressing. The Royals kind of started it a few years ago. They were playing six-inning ballgames for a while. They won a championship that way. I think the rest of the league is trying to copy that. The Yankees tried to do it last year, but it didn't work out for them.
"Nothing is set in stone. As a whole in baseball right now, relievers are a big part of the game. It's making it fun and interesting. There are a lot of good arms out there. It's nice to see them put into games in big situations."
Success across varied relief situations increases Swarzak's value for contenders who could pry him loose before the July 31 non-waiver Trade Deadline. Swarzak wants to stay with the White Sox, and he said he's gotten that point across to the team through light conversations. So he's thinking more about this career achievement over this week -- possibly his last one with the South Siders.
"I sense that I'm a Chicago White Sox now and hopefully for a while. I like it here," Swarzak said. "I like the city of Chicago.
"I'm not in a position to determine where and when I pitch or anything like that, and I don't think I'll ever be that guy. I want to do the best with the opportunity I'm given. Right now, I have the opportunity to pitch for the White Sox."