In the tiebreaker, Fasth allowed just one of three Nashville shooters to score. The 30-year-old improved to 8-0-0 on the season, with four of those victories coming in shootouts.
"We weren't on it right from the start, but we battled our way into the game," Fasth said. "I'm glad that we got away with a win today."
It was the second time in as many games this season that the teams went to a shootout. The first came Jan. 26 in Anaheim in Fasth's NHL debut.
"We played a pretty good game against a pretty good team," Nashville coach Barry Trotz said. "Both times we have played them, it has gone to a shootout. Unfortunately, we haven't been on the positive side yet on that shootout, but did we play well? Yeah, absolutely."
Pekka Rinne stopped one of Anaheim's three shootout attempts. The loss halted Nashville's three-game winning streak.
"Just having that 5-on-5 mentality where you just want to be greasy and get pucks to the net and have people there," Blum said. "Leggy was in the right spot, and I had a good play."
Matt Beleskey evened the score at 1 at 17:25 of the first.
As Beleskey was skating through the slot, Perry hit him with a pass, and Beleskey beat Rinne high for his third goal of the season.
The goal was the first puck to get by Rinne in his last 155:06 of action, a personal best.
Nashville regained the lead at 9:31 when Shea Weber beat Fasth with a wrist shot from the right point.
The Nashville captain's goal was his first of the season and 100th of his NHL career.
Weber's goal came just after Nashville killed off an Anaheim two-man advantage of 1:54 following successive delay-of-game penalties assessed to Rinne and defenseman Hal Gill.
"Usually when you don't score on a 5-on-3, it comes back to haunt you in the end," Anaheim coach Bruce Boudreau said. "This one haunted us right away and they scored. The one thing that our group has been able to do so far is keep their cool and play the way they are supposed to and just keep going and eventually something goes in. Today it was a guy who got his first goal."
Anaheim drew even again at 16:08 of the second on Patrick Maroon's first career goal.
Kyle Palmieri had the puck on the left boards and threw it in front, where Maroon was able to tip the puck past Rinne.
"We were down 2-1, so we knew what we had to do," Maroon said. "We had to create some energy and create offense and throw pucks at the net and go to the net. Obviously, it is really emotional. It is great to get the first one out of the way."
Anaheim concluded its season-long six-game road trip 5-1-0.
Nashville C Colin Wilson played in his 200th career NHL game. He has points in three consecutive games. ... Predators LW Sergei Kostitsyn recorded his 100th career assist. ... After allowing at least one power-play goal against in its last four games, Anaheim did not allow one in Nashville's five opportunities with the man advantage.