Rolston scored in the second extra round of the tiebreaker after Nashville's Paul Kariya squandered a chance to win the game with a penalty shot with 38 seconds left in overtime. After Rolston's goal gave the Wild a 3-2 lead in the shootout, Josef Vasicek couldn't find the net on the Predators' final shot.
Nashville scored three goals in each of the first two periods, but the Wild responded with five goal over the final 40 minutes to force overtime.
David Legwand gave Nashville a 1-0 lead with a breakaway goal at 8:09 of the first period.
Minnesota answered at 10:09 when Wes Walz skated alone through the slot and one-timed a shot between the pads of Chris Mason for a short-handed goal.
Ryan Suter ripped a shot from just inside the blue-line that beat Manny Fernandez to his glove side at 17:38 for a 2-1 Nashville lead.
Dan Hamhuis gave the Predators a two-goal lead less than a minute later, scoring for the first time this season.
Mikko Koivu scored on the power play with a shot from above the right circle at 2:28 of the second period to bring Minnesota within 3-2, but Kimmo Timonen padded the Wild's advantage again with 7:32 left in the frame.
Pascal Dupuis cut the deficit to 4-3 with 6:54 remaining before Nashville's Vernon Fiddler and Steve Sullivan scored goals 2:13 apart.
But the Wild got one back in the final minute of the period when Stephane Veilleux made it 6-4 with his second of the season. Fernandez was then replaced by Niklas Backstrom.
That move wasn't enough to halt Minnesota's comeback.
Pavol Demitra got the Wild within a goal 5:26 into the third, and Brent Burns tied it with a one-timer from the slot at 11:39.
Mason finished with 39 saves in a losing cause.
Notes: Legwand and Sullivan are tied for the team lead in goals with six. Walz has four points (2 goals, 2 assists) in his last three games. Nashville is 4-0 in the second half of back-to-back games this season. ... Nashville's J.P. Dumont had three assists, giving him six in three games against the Wild this season. Demitra has three goals and three assists against Nashville.