The victory was Anaheim's fourth straight and eighth in its last nine games. Nashville's loss halted a three-game winning streak.
It was the second time in as many games this season that the teams went to a shootout.
In the tiebreaker, Anaheim goaltender Viktor Fasth allowed just one of three Nashville shooters to score. He improved to 8-0-0 on the season, with four of those victories coming in shootouts.
Pekka Rinne stopped one of Anaheim's three shootout attempts.
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 defenceman Hal Gill.
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 by Rinne.
Anaheim concluded its season-long six-game road trip 5-1-0.
NOTES: 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.