Nashville took a 1-0 lead midway through the first period after Craig Smith charged past defenseman Ryan Suter to set up Gaustad's goal.
The Wild invested $196 million to bring in Suter and high-scoring forward Zach Parise during the offseason, but the investment hasn't resulted in more offense yet.
Wild fans booed the home team during a 4-1 loss to Vancouver on Thursday, and were restless on Saturday before Clutterbuck tied the game at 1 by tipping in Suter's shot from the point at 12:59 of the second.
Nashville entered the game allowing only 1.81 goals per game, lowest average in the NHL.
Rinne has turned away 112 of the past 115 shots he faced, but Mason got the nod because the Predators will be playing in Chicago on Sunday.
In his second start of the season — both against Minnesota — Mason stopped 30 shots.
Unfortunately for Nashville, Backstrom was just as good on the other end.
Backstrom, who was pulled early in Thursday's loss, is 19-2-4 in games after being removed for non-injury reasons.
The bounce-back game was needed, especially when the Wild had to kill a four-minute high-sticking power play midway through the third period and another two-minute high-sticking call later in the third.
"You want to get back there," Backstrom said. "Every time you get pulled it's not fun. You take it pretty personal. You want to go back out there and make sure you're not letting your teammates down."
NOTES: Gaustad's goal was his first since last Feb. 19. ... Wild forward Dany Heatley has no points in six games after he scored goals in four of five games. ... The Wild have won just two of nine against Nashville.