The Ducks took a 1-0 lead at 17:47 of the first period after Dallas turned the puck over in the corner of their zone.
Anaheim forward Robby Fabbri retrieved the puck and drove through the right circle before it was poked off his stick. The puck went off the skate of two Dallas players before Gauthier skated in and shot it into the net from the slot.
Oettinger made a right pad save on Killorn with the top half of the net exposed while on a power play with 58 seconds left in the second period.
Jamie Benn then had a shorthanded breakaway going the other way, but Gibson made the save before Benn was called for slashing Gibson, giving the Ducks a 5-on-3 power play for 24 seconds.
"He didn't hear a whistle," DeBoer said of the slashing penalty.
Anaheim didn't capitalize on the two-man advantage and the Stars scored shortly after the second penalty ended and during a delayed penalty on Anaheim.
Benn received a pass after exiting the penalty box and he drew a tripping penalty on Zellweger, but before the power play began Sam Steel centered a pass to Blackwell for a redirection that tied it 1-1 at 1:29 of the third.
"They got a lot of really good forwards, a lot of heavy guys, so I thought we did a good job line matching," Zegras said. "When our guys were out there, we were shutting down their top guys. We were trying to at least, and Gibby was making some big saves, which helps."