Weber opened the scoring at 15:52 of the first period.
With Nashville on a power play, Roman Josi sent a pass from above the right faceoff circle to Weber above the left, where he blasted a one-timer over the glove of Flames' goaltender Karri Ramo.
The Predators outshot Calgary 15-10 in the first.
"We turned the puck over in the first period," Calgary coach Bob Hartley said. "We kind of gave them the first period that we didn't want to give them. We had no intensity. If it's not for Karri Ramo, we could have gone on the plane after the first period."
The Predators entered Tuesday with the league's seventh-ranked power play, but had not scored a goal on the man advantage in their last four games.
A night after recording his first-career NHL shutout in Carolina, Ramo got the start Tuesday and made 30 saves.
The Flames drew even with 1:04 remaining in the first.
From behind the Nashville net, Jiri Hudler carried the puck just above the goal line on the left side and slid a pass to an unchecked Stempniak in the lower part of the right faceoff circle. His shot beat Nashville goaltender Carter Hutton for his seventh goal of the season.
Stempniak's goal was his first since Dec. 7, a span of 17 games.
"I thought I did a good job tonight of handling the puck and keeping the forecheck at bay," Hutton said. "I thought we had the edge in the first. I thought (Ramo) made a lot of good saves and for me, I was able to make a couple of saves and handle the puck and get in the game."
Hutton saw his first game action since a disappointing effort in Carolina on Jan. 5 that drew some criticism from Nashville coach Barry Trotz. He recovered nicely Tuesday night, making 22 saves.
"That was a good bounce-back game for him," Trotz said. "I thought we did a really good job of containing Calgary for the most part, especially in the first two periods."
Nashville went in front for good at 55 seconds of the third when Ellis carried the puck in on the left side, cut to the high slot and beat Ramo with a slap shot for his second goal of the season.
"You just keep shooting, I think that's the biggest thing," Ellis said. "Our forwards did a good job of driving the net, and luckily it went in."
Less than two minutes later, Smith gave Nashville a 3-1 lead on another Predators power play.
Seth Jones' slap shot from the center of the blue line hit traffic in front of the Calgary net, and Smith dug the puck out and beat Ramo with a wrist shot to the glove side.
Calgary though they had cut the Nashville lead at 4:17 of the third, but the officials ruled that Mark Giordano had played the puck to a teammate with a high stick.
Stalberg put Nashville ahead 4-1 at 10:52 of the third when his redirection of Ellis' shot got by Ramo, but Backlund made it a two-goal game at 13:07 with a nice move off the right half wall before beating Hutton on the stick side with a wrist shot.
Backlund has a goal in each of his last three games.
"It's nice for personal confidence," Backlund said. "That's what everybody wants to do, and everybody tries to do. I'm happy about it, but at the same time, if we don't win games, I'm not satisfied."
Hudler had two assists in the game. He has assists in three consecutive games. ... Nashville is 17-2-4 when scoring first this season. ... The Predators concluded a five game home stand 2-2-1. . It was the first three-assist game for Legwand this season and the fourth of his career.
Copyright by STATS LLC and The Associated Press
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