Nelson, whose fastball touched 98 mph early in the game, struck out five and walked one while improving his record to 8-2. He has now fanned 96 batters in 901⁄3 innings with Nashville this season.
Luis Jimenez hit a two-run homer and Brennan Boesch added a solo shot for the Bees, but Nelson was in control most of the way.
"I thought he threw the ball well," said Salt Lake manager Keith Johnson. "Obviously he has a power sinker. He has a pretty good slider. His secondary stuff is really good for someone his age. He's going to have a long career in this game."
Nelson also got a little help from the Los Angles Angels.
In the 24-hour span starting Friday morning, the Bees' parent club called up Grant Green and Efren Navarro, leaving Salt Lake without two of its best bats.
In addition, Taylor Lindsey and Matt Long remain on the disabled list, although both made a rehab start with Orem in the Pioneer League on Saturday.
Asked about all the roster moves the Bees have endured this season, Johnson smiled.
"And it will just keep coming," he said. "At the same time, it's opportunity for guys. … As long as those guys play well and the big club continues to win, it's all worth it."
Nashville, which snapped an eight-game losing streak, took a 1-0 lead in the second inning. Hunter Morris led off with a double and, one out later, Jeremy Hermida singled him home.
The Sounds bumped their lead in the fourth on Hermida's two-run homer, but Boesch kept the Bees close in the bottom half of the inning with a long home run to right on a 2-2 pitch from Nelson.
It was Boesch's sixth homer in the last six games.
After Nashville added a run in the fifth on Jeff Bianchi's sacrifice fly, the Sounds broke away with a three-run sixth. Sean Halton and Hermida singled to open the inning before Matt Pagnozzi drilled homer, making it 7-1.
O Nashville snaps an eight-game losing streak with a 7-4 win over Salt Lake.
• Brennan Boesch goes 3-for-4, including a solo homer, for the Bees.