The Ducks killed off a tripping penalty on Ryan Getzlaf 27 seconds into overtime.
Cam Fowler scored, and John Gibson made 22 saves for the Ducks (13-14-4), who had lost two in a row. It was their first win in the past six road games (1-3-2).
"We've been in a lot of close games lately and coming out on top of this one, it's a huge deal for us," Rakell said. "Just the environment, feeling in the room, even though we have been playing well but we lost close games -- at the end of the day we lose -- so being on top of this one feels great."
WATCH: [All Ducks vs. Wild highlights]
Ryan Hartman and Ryan Donato scored and Kaapo Kahkonen made 31 saves for the Wild (14-12-5), who are 5-1-1 in their past seven games and have a point in 12 of their last 13 (8-1-4).
Minnesota trailed 2-0 before rallying to tie the game. The Wild were outshot 14-1 in the first period.
"The whole night was pretty frustrating, quite frankly," Wild coach Bruce Boudreau said. "To start out that first period like that, to get one shot on goal at home, it's pretty embarrassing, and I told the guys that. I said, 'Our first 10 games weren't very good, like with the won-loss record, but nothing compares to what we did in the first period."
Hartman took a wide-angle shot that dribbled past Gibson at 7:59 of the second period to cut the lead to 2-1.
Zach Parise backhanded a pass to Donato for the one-timer top shelf at 2:03 of the third period to tie the game 2-2.
