Roman Josi scored, and Rinne made 22 saves for the Predators (49-16-11), who ended a three-game losing streak.
WATCH: [All Wild vs. Predators highlights]
"I think it was our best game for a while now," Josi said. "I thought we played really well defensively. They didn't have too many chances, and obviously [Rinne] whenever they had a chance was there for us. I thought it was a complete effort from us. It's good to be back on the winning ways."
Eric Staal scored his 40th goal of the season, and Devan Dubnyk made 28 saves for the Wild (42-24-10), who are in third place in the Central Division, three points ahead of the St. Louis Blues.
"The effort, I thought, was great," Minnesota coach Bruce Boudreau said. "To come back against that team in this building, especially after the first, you take [five] penalties … in the second, I thought it showed a lot. It was a great point."
The Wild challenged an apparent game-winning goal by Predators defenseman Ryan Ellis 40 seconds into overtime for offside and were successful.