Cozens pushed it to 4-2 at 13:55 with a power-play goal from the left circle.
"We talked about it with you guys here this morning of how good they are offensively in the League, and they proved that," Wild coach Dean Evason said. "I don't know, maybe we wanted to prove how good we are, but that's not who we are. We traded chances with them and in the end got burnt by it."
Hartman cut it to 4-3 at 18:45, deflecting a cross-ice pass from Gaudreau out of midair at the side of the net.
Marcus Foligno scored 26 seconds into the third period to tie it 4-4 when a pass from Eriksson Ek redirected in off his skate at the edge of the crease.
Eriksson Ek then put the Wild ahead 5-4 at 7:48, scoring glove side on the power play.
"That one's not on [Fleury], he played great, and give credit to that team," Foligno said. "They do swarm and come in a group of five a lot of the game. We battled hard, but I think the start for this game [was] a little bit sleepy on our end.
"I think we knew the offensive talent that they had and the way they play. I think we could've done a better job in the first to understand the style of play, but they definitely swarmed us and kept us on our heels, and it took us a while to get to our game. … All in all, you've got a lead with a couple of minutes, you've got to bear down and didn't do it. Overtime was nuts, so it's just one of those things you'd like it to not go to overtime."