Mikhail Sergachev tied it 2-2 at 10:44 of the second period with a shot from the point that deflected off of the stick of Arizona forward Lawson Crouse and knuckled past Vejmelka.
"It was a tough start, going down 2-0 and then going on the PK," Hedman said. "Great kill by us and we gained some momentum. 'Pointer' worked his magic, batting it out of there, and got us going. That was big for us."
Stamkos put Tampa Bay ahead 3-2 at 11:12 with a one-timer from the left circle on a power play.
Bjugstad tied it 3-3 at 12:44 when he capitalized on a turnover by Sergachev and scored with a wrist shot from the left circle.
"We had the lead and the game turned, it was toe-to-toe," Tourigny said. "I don't think there were many swings in the momentum. It was a 60-minute, hard-fought battle on both sides."