Smith was able to convert on a breakaway after taking a pass from Mike Ribeiro. The goal was Smith's 14th of the season.
"I think in overtime you have to be patient," Smith said. "[Nashville goaltender Carter Hutton] made some great saves there at the end. We were scrambling a little bit to hang on to the puck and make a tape-to-tape pass, but [Ribeiro] got the puck, put his head up and made a nice saucer pass through the middle."
Colorado defenseman Nick Holden tied the game 3-3 with 36 seconds remaining in the third period while the Avalanche had six skaters. Holden was left alone in front of Hutton and placed the puck under the crossbar.
"That's certainly not what we want to do at the end of the game," Predators coach Peter Laviolette said. "I thought our guys played really hard in the second period and the third period. I liked the fact that we scored early in the third period and continued to press for the entire time. Those last two or three minutes, you're trying to close out a game and they've got the extra attacker. There's more bodies for them on the ice than we can cover."
Filip Forsberg gave Nashville a 3-2 lead 32 seconds into the third period. He took a wrist shot that Colorado goaltender Semyon Varlamov paddled away, but Forsberg got to the rebound and was able to convert from a tough angle. The goal was Forsberg's 16th of the season, and his 41st point in 46 games, best among NHL rookies.
"I was just trying to get the first shot on net, and then obviously the rebound came right out," Forsberg said. "I just tried to get it back at the net. I didn't really see if it went in or not, but I kind of heard everyone celebrating."
Nashville (31-10-5) is 19-2-1 for the best home record in the NHL.
"It's just about hard work," Forsberg said. "We came out pretty good this game and then kind of dropped a little bit, but then we took it back in the second and third. I guess that's how we want to have them. We want to be really tough to play against at home especially, and I think we've been doing a pretty good job of that so far."
Nashville outshot Colorado 39-24 and dominated the territorial play in the second half of the game.
"We weren't really happy with the first period," Smith said. "I think as it went on, we kind of woke up a little bit and got back to our old selves. Dust the shoulders off a little bit and get back out there."
Cody McLeod gave the Avalanche a 1-0 lead at 11:52 of the first period when he redirected a shot past Hutton. Defenseman Zach Redmond took a shot from the point that McLeod tipped in front of the net.
Nashville defenseman Roman Josi tied the game 1-1 at 3:33 of the second period on a one-timer from the point. Josi one-timed a pass from defense partner Shea Weber that made its way through traffic and past Varlamov.
"[Josi] and Weber are basically the perfect fit for defensemen," Forsberg said. "Weber is strong defensively, and Roman is just flying around. It feels like they don't even look at each other when they pass the puck to each other. It's just awesome to have them on our team."
Colin Wilson gave the Predators a 2-1 lead at 13:21 of the second period with a power-play goal. Defenseman Seth Jones took a shot from the point that missed the net, but Wilson was able to corral the rebound and shoot it past Varlamov. It was Wilson's 19th point in the past 15 games.
Colorado forward Jarome Iginla tied the game 2-2 at 16:05 of the second period on a one-timer from the faceoff circle. Forward Matt Duchene delivered a drop pass that Iginla was able to place over Hutton's shoulder for his 15th goal of the season.
"This year I've gotten an opportunity up front," Iginla said. "There's so many dynamic players, it's getting some good looks. When you play with [Alex Tanguay], you get even more good looks, and I really enjoy playing with him and [Duchene]."
Ribeiro had three assists and has 32 this season. He passed Forsberg for the Predators' lead in points with 42 in 46 games.
"I don't even know how he does it sometimes," Forsberg said. "You saw the last pass he made to Smith there, a saucer right on the tape and game-winner right there. That's the stuff he does all the time. It's awesome for us."
Colorado (20-18-11) hosts Nashville on Friday, after the Predators play at the St. Louis Blues on Thursday.