Leonard was promoted from Milwaukee of the American Hockey League earlier in the day.
"It felt awesome to play tonight," Leonard said. "It was a quick last 24 hours, and everything happened quick, but the guys were great helping me out. It was awesome being able to find a way to win tonight with big saves and a lot of shot blocks as well. It was a full team effort, and it was a lot of fun."
Matt Duchene also scored, Roman Josi had two assists, and Luke Evangelista had his first NHL point for the Predators (30-23-6), who have won four of their past five games.
"The last three games for sure, we've played with some swagger in our game, which is nice to see," Nashville coach John Hynes said. "I think guys are making some good plays and feeling confident with the puck, which is important. I think we're playing really fast and the thing that drives all of it is throughout the lineup, the competitive level and work ethic is at the standard it needs to be to give yourself a chance to win every night."