Jake Oettinger made 31 saves in his fourth NHL shutout for the Stars (14-6-4). Miro Heiskanen had three assists, and Roope Hintz had two assists.
Anthony Stolarz made 36 saves and stopped Jamie Benn on a penalty shot in the third period for the Ducks (6-16-2), who have lost four straight.
"It's a good team, they're well balanced," Anaheim forward Simon Benoit said. "They bring that puck deep, they go to the net. When we give them a little bit of space, they just take it. I think that's what they did. We let off that hook a little bit, and they just capitalized on those [mistakes] we have."
Robertson became the first 20-goal scorer in the NHL this season when his wraparound went in off the stick of Ducks defenseman Urho Vaakanainen to give the Stars a 1-0 lead at 4:13 of the first period.
He scored his second of the game on the power play to make it 2-0 at 7:43 of the second period when his shot from the high slot went off the left post and in.
"He's our first-line guy, and he could take all the optionals and show up last on the ice and stuff, but he's the exact opposite," Oettinger said. "He wants to do the goalie drills every day. He likes to have fun with us, but I think he's doing it because he wants to learn how to score. Obviously, it's working, so he needs to keep doing that."
Joel Kiviranta made it 3-0 at 11:09 when the rebound of Heiskanen's shot landed in his chest and he snapped it past Stolarz.
"To a man, we should all be embarrassed with the effort," Ducks forward Derek Grant said. "Once they got their second and third, it's like we all just sat back and didn't want to win. You can't play like that in this league, you're going to get burned. Guys need to be [ticked] off, they need to be embarrassed. It's not fun for anyone, but you've got to find a way to take that and use it for fuel."