Dustin Penner and Corey Perry scored for Anaheim in the second period, and Scott Niedermayer added a power-play goal in the third. Niedermayer's goal proved the difference, as Minnesota's Pierre-Marc Bouchard scored his second goal of the game to close the gap to 3-2 with 53 seconds remaining.
Bouchard made it close with his third goal of the season after a defensive lapse by the Ducks allowed him to beat Giguere at close range.
While Giguere faced 26 shots, the Ducks got off 38 at Niklas Backstrom.
The Ducks' season-opening run of 16 games without a loss in regulation ended in a 3-0 loss to the Flames in Calgary on Friday night. By earning at least a point in the first 16 games, the Ducks set an NHL record.
The game against the Wild was the third between the teams since Oct. 20 in Anaheim, when the Ducks won 2-1 behind Giguere's 27 saves. The Wild took a 3-2 shootout victory in Minnesota the following week.
After Anaheim took a 2-0 lead in the second period this time, Bouchard cut the deficit when he lofted a backhander past Giguere at 7:55 of the period.
Niedermayer gave Anaheim a two-goal pad with his fifth goal, at 5:11 of the third. With Backstrom sprawled to the left of the goal after blocking a shot, Niedermayer slammed a shot from the right circle into the open net.
Ryan Getzlaf picked up his second assist of the game, and Teemu Selanne also got an assist, his team-high 16th, on Niedermayer's goal.
Following a scoreless first period, Perry gave the Ducks a 2-0 lead at 5:41 of the second. Backstrom blocked Shane O'Brien's slap shot, but the puck trickled behind the goalie in the crease and Perry poked it into the net for his fifth goal.
Penner opened the scoring with his sixth goal, at 1:07 of the second period. Penner brought the puck from behind the goal line and beat Backstrom with a 15-footer from the right side on a difficult angle.
Anaheim outshot Minnesota 13-8 in the first period, with Giguere gloving Bouchard's shot on a breakaway, and Backstrom blocking a point-blank shot by Samuel Pahlsson to keep it scoreless.
Notes: Anaheim LW Chris Kunitz, who injured his left hand Friday night at Calgary, did not play against Minnesota and is scheduled to have an MRI on Monday. He has eight goals and five assists this season. ... The previous record streak to start the season without a loss was by the 1984-85 Edmonton Oilers (12-0-3). ... The Wild were coming off a 3-2 shootout win over the Los Angeles Kings on Saturday night when goaltender Manny Fernandez notched his 100th career win by making 30 saves. ... Minnesota RW Marian Gaborik missed his ninth consecutive game because of a groin injury.