Miller scored his first career shootout goal in the eighth round and the Red Wings survived Minnesota's third-period rally to beat the Wild 4-3 on Thursday night.
"You've got to get your first one sometime, and it feels pretty good," Miller said.
Detroit goalie Jimmy Howard stopped six shots in the shootout and denied Eric Belanger in the eighth round to set up Miller's winner. Miller beat Niklas Backstrom, who entered the game after the first period in place of injured starter Josh Harding.
Detroit had lost its last three shootouts and needed to win this one to avoid what would've been a frustrating loss. Before Thursday, Detroit's last shootout win was Nov. 28 against St. Louis. The Red Wings are 4-5 overall in shootouts this season.
Patrick Eaves gave the Red Wings a 2-1 advantage early in the third. Miller appeared to finish off the Wild when he took a pass from Derek Meech and made it 3-1 midway through the period, but the Wild continued their third-period success at home.
Havlat's unassisted goal at 13:48 held up after a review, and Brunette tied it a minute later, beating Howard after controlling a bouncing puck.
Minnesota has outscored opponents 11-3 in the third in its last five home games.
"It's amazing that we almost came back but you can't play hockey like that," Havlat said. "It causes you to lose your power and strength for the next games too."
Todd Bertuzzi also scored for Detroit.
Harding, Minnesota's backup , was making his first home start in over a year. He left with a lower-body injury after the first period and was replaced by Backstrom, the regular starter, to start the second. Wild coach Todd Richards said he thought Harding injured his left hip while making two saves on a breakaway led by Pavel Datsyuk and Eaves early in the first.
Backstrom, an All-Star last season, has struggled at times in Minnesota's new offensive-minded system. Backstrom had given up three goals or more in eight of his last 10 starts before Thursday.
Backstrom looked sharp to start the second and the Wild tied the game at 12:34 on fluky goal from Robbie Earl, who cleared waivers earlier this week but remains with the team. A shot from Shane Hnidy bounced off the side of the net, off the end boards, back over the net, and off of Earl's chest as he crashed the net.
The goal stood after a replay review to determine if the net became dislodged before the puck crossed the line.
"It was like that Magic Johnson-Larry Bird McDonald's commercial back in the day," Howard said. "Off the backboard, off the glass, nothing but net."
Detroit continued a trend of jumping on the Wild in the first period. Bertuzzi forced Kim Johnsson to turn the puck over and scored on a breakaway 7:52 into the first.
The Wild have been outshot 51-16 in the first period of its last four games and have allowed the first goal in four straight.
The Wild have lost four straight and have struggled out of the gate in each of the losses. Richards said he's working to fix the problem.
"I wish I had the answer because if I had the answer it would be corrected," he said. "We're trying different things but we haven't found the right way to go about it yet."
Howard was back in goal for Detroit. The rookie had started 12 straight in place of veteran Chris Osgood before Osgood got the nod Tuesday night in a 3-2 loss at Washington.
Howard stopped 27 shots, including eight in overtime.
Giving up a two-goal lead in the third period is never good, but Detroit coach Mike Babcock was pleased with his team's play after getting two points.
"They scored two goals -- one off the glass tonight and then one with a pop fly that he grabbed and shot in our net," he said. "Those were pretty fortunate bounces. Even though we gave up the two-goal lead it wasn't like they came out and tilted the rink."
Defenseman Brent Burns returned for the Wild after missing the last 29 games because of a concussion. He played about 14 minutes.
Petr Sykora also cleared waivers for the Wild this earlier this week. The 33-year-old forward won't return to the team and the Wild will try to trade him. Sykora missed nearly two months with a concussion and had two goals and one assist in 14 games. ... Bertuzzi has scored against the Wild with four teams. ... Harding had stopped 73 of the last 75 shots he's faced before leaving. It was Backstrom's first relief appearance since Nov. 16, 2007, at Vancouver.