Josh Smith added 21 points, five rebounds and three blocks for the Hawks, who have their longest winning streak since they won four in a row from Nov. 3-10.
"We've kind of got a feel for what each other likes to do," said Smith, who has buoyed his team with a surprisingly early return from hernia surgery. "It's good to be back in the paint, changing shots so it is not just easy lay-ups."
"Josh Smith, Joe Johnson, Josh Childress and Marvin Williams, this is the third time that they have had the chance to play together all season long and it makes a big difference," Woodson said.
The Hawks were aided by an uncharacteristically flat performance from Wolves star Kevin Garnett, who just barely kept his long double-figure scoring streak alive.
One of the most intense and focused players in the league, Garnett seemed a little distant in this one. He missed his first seven shots and managed just 13 points on 6-of-15 shooting, scoring at least 10 points in his 359th straight game, the eighth-longest streak in NBA history.
Mark Blount led the Wolves with 16 points and 11 rebounds, but Minnesota was perhaps a little tired after playing four overtime games this month and had no answer for the Hawks' athleticism and Johnson's smooth shooting stroke.
"I was very disappointed in our approach and our sense of urgency," Minnesota coach Dwane Casey said. "We're not in a position in our program yet to come out and wait until we get behind to play."
Johnson wasted no time taking control, scoring 13 points on 6-of-8 shooting in the first quarter.
He didn't stop there.
The sixth-year veteran, one of the Hawks' elder statesmen, scored 11 more, including three 3-pointers, in the second quarter.
"Single coverage makes life easier," Johnson said. "You get good looks."
He hit two in a row from long range to jump-start a 10-0 run midway through the second quarter that gave Atlanta a 15-point lead. He made 12 of 19 shots for the game and Atlanta shot 54 percent.
"Joe Johnson is one of the most underrated players in the league," Garnett said. "It seems like he made every shot."
And the Hawks were just as impressive on the defensive end, swarming around Garnett to make every shot tough for him.
Garnett was double-teamed, sometimes even triple-teamed, every time he touched the ball. He appeared to grow frustrated as his struggles continued, forcing jumpers and flailing toward the rim, a rare sight for the 12-year veteran.
Even the shots he did make didn't come easy. After his first six misses, Garnett appeared to have a wide open layup underneath, but Childress came from the weak side to swat it away. Garnett corralled the rebound and put it back for his first basket.
And his second field goal came on an ugly bank shot in the lane that clearly was unintentional.
"They took us to the wood shed and used every piece of kindling they had," Casey said.
It was Atlanta's first victory at Target Center in more than nine years. The Hawks had lost eight in a row here, dating to April 12, 1997. ... Claxton missed his third straight game with a sore left knee. ... Wolves F Marko Jaric was out again with a sprained right thumb.