Andrew Cogliano scored in regulation for Anaheim, which has won five in a row and gone nine games without a regulation defeat (7-0-2).
Bobby Ryan also scored in the shootout, while Hiller denied both Dallas shooters he faced.
Ray Whitney, in his second game back after missing 16 due to a broken foot, scored the only goal for the Stars.
Kari Lehtonen, making his sixth straight start, had 29 saves.
Dallas was missing centre Jamie Benn, who sat out his second straight game with a wrist injury, but had numerous opportunities that they didn't capitalize on.
With the game tied at 1 midway through the second period, the Stars had back-to-back opportunities to take the lead.
After entering the Anaheim zone on the rush, Whitney rolled the puck in on goal, but Hiller left a rebound in front.
Just 38 seconds later, it looked as if rookie Reilly Smith put Dallas ahead, but his in-close chip shot over Hiller's blocker ricocheted off the right post, pinged off the crossbar and bounced out behind Hiller.
An Anaheim turnover gave Dallas another breakaway with 6:21 left in the second, but Hiller made a nice save on Eric Nystrom's wrist shot.
Lehtonen made a nice stop on a Ducks power play with 2:17 remaining when he slid across the crease to stop Francois Beauchemin's one-timer from the right side of the crease.
With Dallas on the power play, Lehtonen came up with another big save 48 seconds into the final period when he slid across the crease to get his pad on rookie Emerson Etem's one-timer from the right circle.
The Stars appeared to take the lead again with 3:25 left in regulation, while on a power play, when Jaromir Jagr swiped a shot from the left side of the crease, while falling, past Hiller.
The horn sounded, Dallas players celebrated, but the officials waved it off and play continued for another 33 seconds before there was a video review. Replays clearly showed the puck hitting the right post and crossbar and never crossing the goal line.
The Stars opened the scoring with Whitney getting his third goal of the season 12:35 into the opening period. After Alex Goligoski's wrist shot hit Loui Eriksson's skate in front, Whitney found the loose puck first, sending a backhand over Hiller's glove.
Hiller made a nice save after Whitney's cross-ice backhand pass on a 2-on-1 rush led to Eriksson's point-blank wrist shot 2:47 into the second.
Dallas had two good opportunities to seize a two-goal lead, but was unable to cash in when Smith sped in on a breakaway at 6:32, but his wrist shot sailed over the crossbar.
Just 12 seconds later, the Stars drove back into the Anaheim zone on a 2-on-1 rush that led to Roy's one-timer from the right faceoff circle.
Less than a minute later, the Ducks tied it on Cogliano's 10th goal of the season.
After receiving a nice backhand pass from Peter Holland, Lehtonen made a great save on Cogliano's one-timer from in front, but Cogliano managed to jam home the loose rebound at 7:28.
Notes: Anaheim G Viktor Fasth sat out his second straight game with an upper-body injury and is listed day to day. Jeff Deslauriers, who was recalled from ECHL Fort Wayne on Tuesday, served as Hiller's backup. ... Dallas LW Brenden Morrow returned to the lineup after a two-game absence due to a groin injury. ... Hiller entered the night with a 6-0-1 record with a 1.96 goals-against average and a .931 save percentage in his past seven outings.