Dallas is 6-0 against Los Angeles this season, outscoring the Kings 23-11. The Stars were 1-4-1 overall in their previous six games.
"We needed those points," Stars coach Dave Tippett said. "There was a good amount of desperation and work ethic. We did exactly what we needed to do and got a contribution from everybody."
"This was a big confidence-builder for us," Daley said. "We had a sense of urgency."
Michael Cammalleri had a power-play goal for Los Angeles, which has lost four in a row overall.
Kings coach Marc Crawford picked Brust over rookie Yutaka Fukufuji, who became the first Japanese-born player to play in an NHL game when he came on in relief of Brust in a 6-5 loss at St. Louis on Saturday.
Stars center Jeff Halpern was a late scratch after sustaining a lower-body injury in warmups, and defenseman Philippe Boucher did not return to the ice after the first period because of an upper-body injury.
Dallas remained without five other regulars, including top offensive threat Mike Modano, who missed his 19th consecutive game with a hip/groin injury.
The Stars have been forced to go with rookies and inexperienced players, with mixed results.
"We have had a lot of lineup changes lately and I thought the guys played very well," Hagman said. "We'd lost three in a row and although it's not the end of the world, it makes it harder on us each day."
Dallas scored its 10th power-play goal in 31 chances against Los Angeles this season when Barnes connected for his sixth goal of the season on a close-range backhander at 16:01 of the opening period.
Hagman made it 2-0 on Dallas' second power play, converting Jussi Jokinen's setup at the eight-minute mark of the second period for Hagman's 10th goal of the season.
Cammalleri cut the deficit to 2-1 at 9:43 of the second with his 18th goal of the season. But Daley's goal from the slot at 6:56 of the final period -- Daley's second of the season and sixth of his career -- extended Dallas' lead to 3-1.
"I thought the first two periods we played well, but we got outworked in the third period," Kings defenseman Aaron Miller said. "We were really outplayed in the third. If we don't outwork teams, we're not going to win. That goal early in the third period really hurt us."
Dallas dominated the third period, holding a 15-7 shots advantage with the help of six power plays.
"We were in our end most of the third period, and it's tough to play that way," Los Angeles defenseman Oleg Tverdovsky said.
- Dallas has outscored Los Angeles 11-0 in the first period this season.
- The Stars played a rare matinee on Martin Luther King Day. The crowd was announced at 17,992, but there were plenty of empty seats as fans stayed home because of icy driving conditions.
- Hagman matched his career high for goals.
- Barnes has three goals and two assists against the Kings this season.