Randy Robitaille scored for Minnesota, which was outshot 37-17.
The Wild are one of the few NHL teams that still tries to clog the neutral zone and generate offense by waiting for opponents to turn over the puck. The league's new rules include a crackdown on holding, obstruction and contact with the stick.
That has made the trap -- used in some form by many teams before the lockout -- almost obsolete.
"That's the way they play," Legace said. "They just sit back and wait and capitalize on your mistakes."
Shanahan and Williams connected to give Detroit a 2-0 lead in the second period when the Red Wings held a 20-4 shots advantage.
"It took us the first period to figure out how to attack it," Detroit coach Mike Babcock said. "But we adjusted and I think we were absolutely dominant in the second period."
Babcock said a key was that Detroit defensemen moved the puck quickly and didn't give Minnesota time to set up.
"This is a team that controls the puck well," Wild coach Jacques Lemaire said. "You can't allow any mistakes here."
Shanahan opened the scoring at 7:28 with his 10th goal. He put in a rebound from the bottom of the left circle off a shot from the point by Jiri Fischer.
Williams' power-play goal at 9:26 made it 2-0. After moving in from the right point, Williams took a pass from Henrik Zetterberg and beat Manny Fernandez with a wrist shot from the bottom of the right circle, high to the short side.
"Just trying to sneak down the boards on the power play," Williams said. "Hank [Zetterberg] made a great pass and I just tried to get it on goal up high."
Robitaille made it 2-1 with a power-play goal 8:26 into the third period. His shot from along the goal line to Legace's left, bounced off the inside of one of the Detroit goaltender's pads and into the net.
Legace thought he should have stopped the shot.
Franzen restored the Red Wings' two-goal margin with 8:18 remaining when he put in the rebound of Mikael Samuelsson's shot.
Detroit outshot Minnesota 10-6 in the scoreless first period.
"We gave them too much respect at the beginning and it cost us," Robitaille said.
Zetterberg has at least one point in 13 of Detroit's 18 games. ... Minnesota center Marc Chouinard (shoulder) and defenseman Andrei Zyuzin (strained knee) both sat out. ... Detroit center Pavel Datsyuk had his 10-game point streak snapped.