"I'm not sure why that is, but we've been playing a very patient game on the road," defenseman Niklas Lidstrom said.
Detroit got it done on defense after allowing four goals in each of its past four games.
"We kept it real simple in getting pucks out. We said when there's no play, just get it out of the zone," defenseman Brad Stuart said. "We did a lot better job with that."
The Wild had just four first-period shots and didn't get its first of the second until nearly 5 minutes had elapsed. It had just five in the period and 17 in the game.
Minnesota was booed off the ice after the first two periods.
"We're frustrated," center Matt Cullen said. "We've been playing some good hockey, and to take a step back like that, especially in front of a great crowd, that's all fair game. We should have probably been booing ourselves."
Brent Burns had a late power-play goal for Minnesota, which lost for the first time in four games.
The Red Wings recorded a pair of soft goals in the opening 9 minutes and were in control throughout.
From below the goal line on the left side, Zetterberg bounced a rebound off goaltender Niklas Backstrom and into the Wild net at 4:38. Cleary scored at 8:59 when his shot from below the right goal line deflected off Backstrom's skate and across the line.
Backstrom allowed just three goals during Minnesota's three-game winning streak.
"You just try to get the pucks in there for rebounds and they went in for us," Lidstrom said. "We'll take those goals, especially coming off the two-day break."
It was a long day for the Red Wings, who had a morning skate in St. Paul and played a game that started one hour earlier than normal. They were scheduled to fly to Colorado after the game and face the Avalanche on Monday.
"The way the last couple days went with spending the 24th and 25th at home having Christmas, and having a real early flight this morning to get here, we'll take this all day long," Draper said. "We just wanted to play a real simple game."
Draper made it 3-0 early in the second period when he fought off Clayton Stoner and tipped a harmless looking wrist shot by Niklas Kronwall past Backstrom. It was the first goal of the season for Draper, who missed the team's first 22 games following hernia surgery.
"When you score the first two, three goals, it certainly makes it a lot easier to go and play," Draper said. "We did a real good job in the neutral zone; any chance we got we got the puck in deep and we got on the forecheck and certainly got them on their heels."
Holmstrom outmuscled Matt Cullen at the right post for a rebound goal at 13:50 of the second period, after Zetterberg drove the lane and fought off two Minnesota defensemen to get off a shot.
Detroit goaltender Jimmy Howard tied Montreal's Carey Price for the league lead with his 19th win. He was rarely tested until Burns poked the puck past him with 3:40 to play.
"They just kept coming. It took us a long time just to put up a fight," Minnesota center John Madden said.
Fifteen of 18 Wild skaters were minus-1. The other three were even. ... Kronwall had two assists to equal his total from the previous 12 games. ... Zetterberg has seven points, including three goals, in his last three games. ... Johan Franzen has five goals and three assists in his last four games against Minnesota. ... Wild D Nick Schultz, who missed his third straight game with a possible concussion, could play Monday in Columbus. ... Attendance for the game was 19,227, the fifth-largest regular-season crowd in Wild history.
Copyright by STATS LLC and The Associated Press
SAINT PAUL, MN - DECEMBER 26: Matt Cullen #7 of the Minnesota Wild stretches prior to the game against the Detroit...