With Suter serving a roughing penalty, Predators captain Shea Weber sent a pass to Sergei Kostitsyn on the left side. Kostitsyn then slid a cross-ice pass to Legwand for a shot that sailed over Backstrom's glove.
"We got a point," Weber said. "It was a nice win for us last night. We were trying to get something going here before we get going on the road, but now we're forced to do something on the road when we go away."
Backstrom made 24 saves through overtime. Rinne stopped 28 shots before the shootout.
Nashville fell to 2-5 in shootouts this season.
"We played a real good Minnesota team that got a chance to scout us last night," Nashville coach Barry Trotz said. "They were pretty ready and on top of their game in a lot of ways."
For the second time in two nights, the Predators sustained an injury to a forward. This time it was leading scorer Colin Wilson.
Early in the second period, Wilson was checked hard into the end boards in the Minnesota zone by defenseman Justin Falk. Wilson left the game and didn't return because of an upper body injury.
Wilson leads the Predators with seven goals and 12 assists. Trotz said there wouldn't be an update on Wilson's condition until Monday.
On Friday, the Predators lost forward Patric Hornqvist to an upper body injury. There is no timetable for his return. Hornqvist led the Predators in goals in two of the previous three seasons.
NOTES: Minnesota RW Cal Clutterbuck returned to the lineup after missing six games because of a lower body injury. ... Suter has a six-game point streak. ... Weber has five assists in three games.