Mike Ribeiro recorded a goal and two assists, while James Neal and Brad Richards each scored and added a helper for Dallas, which was coming off an 8-2 victory over Toronto and has won four of five overall. Marty Turco was shaky in net, but made 19 saves to pick up the win.
"Winning is addictive and losing is addictive," Richards said. "Right now we're winning and it's becoming addictive. You've got to turn it your way and we've turned it our way with hard work."
Brendan Morrison lit the lamp twice and Chris Kunitz netted Anaheim's other goal as the club dropped its third straight game, each coming by a 4-3 score. Giguere was tested throughout the night, logging 34 saves.
With 3:21 remaining in regulation, the Ducks took a 3-2 lead, as Bobby Ryan faked going around the net with the puck, wheeled and fired a pass to Morrison, who one-timed the disc past an unsuspecting Turco.
The Stars emptied the net with roughly 1:20 left and the strategy paid off, as Neal's cross-ice pass through traffic hit Ribeiro at the left circle before being relocated over Giguere's right shoulder and into the net for a tie score with 64 seconds remaining.
Ryan Getzlaf was whistled for slashing 1:33 into overtime and just as the Stars' man-advantage was winding down, Richards fed Modano just over the left circle and the veteran unloaded a blast that proved to be the game-winner one second after Getzlaf was released from the penalty box.
"We are showing a desperation for points, but we are giving away too many points," Stars head coach Dave Tippett said. "Great to see us come back and send it to overtime, and Modano had a great shot at the end, but we were sloppy and messy...We'll take the points though."
Kunitz slipped a goal past Turco 6:20 into the game for the only tally of the opening period.
Richards tied things 8 1/2 minutes into the middle period by going top shelf on Giguere's glove side after an Anaheim turnover in the defensive zone.
The teams traded similar power play goals in the third stanza. Neal put the Stars ahead 1:06 into the frame on a rebounded shot from the point, and Morrison answered just over seven minutes later with a quick backhander after Turco couldn't control a slap shot from the blueline.
Modano also added an assist...Dallas has won four straight at home...The Ducks are 1-3 on a five-game road trip...The Ducks had won two straight in Dallas.