Fasth has opened his NHL career by going 8-0-0, including four shootout victories. He made 32 saves before stopping two of three attempts in the tiebreaker. Ray Emery of the Chicago Blackhawks holds the record by winning his first nine NHL starts.
The Ducks, now 11-2-1, have won four in a row and eight out of nine, including a franchise-best five of six on this just-finished road trip.
"[Fasth is] playing exceptionally well, and guys are playing well in front of him," Perry said.
Anaheim had to kill off a Nashville power play at the start of overtime due to a bench minor for too many men near the end of regulation. Mike Fisher had a redirection that went off the crossbar.
Rinne allowed his first goal in almost eight periods, and Predators defenseman Shea Weber scored his first of the season.
Weber's goal, the 100th of his career, was a long shot from the right-wing boards that likely deflected off Anaheim defenseman Luca Sbisa and went past Fasth at 9:31 of the second period. It came 30 seconds after Nashville killed off a 5-on-3 power play during which it allowed just one shot on goal.
For Weber, who signed a 14-year, $110 million contract this offseason, it was his third point and first regular-season goal since March 31, 2012.
Ducks forward Patrick Maroon scored his first NHL goal to tie the game 2-2 at 16:08 of the second. Kyle Palmieri skated out from behind the Nashville net, up the left-wing boards, turned and took a shot that Maroon -- a 24-year-old picked in the 2007 draft -- deflected past Rinne.
Anaheim's Matt Beleskey tied the game 1-1 when his hard shot from the slot off a behind-the-net feed from Perry beat Rinne glove side with 2:35 left in the first. That ended Rinne's scoreless streak at 155:06. He had not allowed a goal since Chicago's Patrick Kane scored 4:27 into the third period of a 3-0 loss to the Blackhawks on Feb. 10. Rinne shut out the San Jose Sharks and Phoenix Coyotes in his next two starts.
Rinne made 20 saves Saturday.