The 24-year-old Englishman attacked on the climb heading towards town, in the northwest Galicia region, and soloed towards his first WorldTour and Grand Tour victory.
"He's not a future champion, but he's one real champion now," said Esteban Chaves. "We are happy for him and we'll enjoy this."
Chaves is racing for the Vuelta's overall after winning two stages and placing fifth last year, and finishing second overall in the Giro d'Italia this May. However, he wanted Yates to have his chance and the team pushed hard through the stage to create a launch pad.
"Today, Simon had the opportunity," Chaves continued. "He's a real champion and he deserves this. He's really consistent, he's disciplined, he sacrifices himself. He's a good boy."
Yates's twin brother Adam impressed at the Tour de France by riding as high as second behind Chris Froome (Sky), and placing fourth overall to take home the white young rider jersey.
How well do the Yates brothers know each other?
Simon Yates won stages in the Tour de L'Avenir and in the Tour of Britain before officially turning professional with his bother in Australian team Orica. Last year, he had a string of top placings: fifth overall in País Vasco, sixth in the Tour de Romandie and fifth in the Critérium du Dauphiné. He just lacked a big win.
"It was a really awesome win for him. It was amazing how he just took control of it. It was classy and I haven't seen a win like that in a long time," said former professional and team sports director, Julian Dean.
"I think with him, and I saw this with Adam at the Tour as well, they're very composed characters. They just thrive on racing their bikes. They just want to be able to do stuff like what Simon did today. That's his biggest motivation, to be able to pull off moves like today.
"How he does it, I don't know. I don't think they don't get too distracted by outside influences. They just focus on training and they are not big characters on social media, they don't care. They just want to focus on racing their bikes well."
One by one, Yates's team-mate arrived in the small town near Ourense, where the Vuelta took off on its three-week journey on Saturday. Chaves was there immediately to congratulate Yates and others, each in the team's white and blue kit, trickled in afterwards.
"It's huge," Jens Keukeleire said. "A Grand Tour stage win is always special, especially if it's your first.
"They are both some of the biggest talents I've ever raced with. It is their ability to do what they are doing now at such a young age. I remember I was there with Adam when he won the San Sebastián classic last year. To do that at such a young age is really amazing."
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