Leon Draisaitl scored 1:34 into the third period before the Oilers pulled Connor Ingram for the extra attacker to make it six-on-five with over four minutes left in regulation, cutting the lead to one on Connor McDavid's 34th goal of the season before Evan Bouchard sent it to overtime with 59 seconds remaining.
Zach Hyman put home the game-winner 1:06 into overtime to secure the 4-3 win and his team's first three-game win streak this season.
"I think nobody really wanted to talk about it, and it's good that we got that third one out of the way," Kapanen said. "So I'm sure guys have been thinking about it a little bit throughout the year, but hopefully, from now on, we can just keep playing well and then extend this win streak."
Hyman said it was a good reminder for their group to keep pushing when the result doesn't look like it's going to go their way.
"I think this group is pretty confident that no matter how we're playing, we can climb out of a hole," Hyman said. "You obviously wanna start out great, you wanna be in the lead, you wanna do all the right things, but hockey sometimes doesn't work out like that. So it's good for us to win a game like this.
"You don't want to have it happen often, but every once in a while, it's good for your team."
The Oilers will wrap up their season series with the Wild after losing the first two meetings – a 1-0 shutout on Dec. 2 at Rogers Place on a Jonas Brodin goal
and 33 saves from Jesper Wallstedt, followed a few weeks later on Dec. 20 with a 5-2 defeat at Grand Casino Arena behind a pair from Matt Boldy.
Winning 4-1 at home over Calgary on Thursday, the Wild have won two straight and sit second in the NHL's Central Division at 31-14-10.