Although Carolina was outshot 22-10 in the final two periods, a timely goal by Brind'Amour and a strong effort by Leighton gave the Hurricanes their first victory on a six-game trip that began with a 4-3 overtime loss at Los Angeles on Friday night.
"It wasn't necessarily pretty on our end. They probably had the majority of the chances, but Michael really came up big for us," Carolina forward Matt Cullen said. "Sometimes you need that on the road. It's a good confidence builder for us."
After grabbing an early lead the Hurricanes saw the Ducks control play and draw even in the second period on a goal by Francois Beauchemin.
But Brind'Amour quickly responded, gathering a puck in the slot and calmly swiping around Anaheim goalie Jean-Sebastien Giguere to score his third goal of the season, giving the Hurricanes a 2-1 advantage after two periods. Staal added an empty-net goal in the final minute and Leighton did the rest.
"That was pretty ugly really," Brind'Amour said. "We relied on our goalie way too much."
Leighton was acrobatic in the third period when the Ducks tried to convert on three power-play opportunities, including an extended 5-on-3 advantage.
"We were just sitting back and watching them later in the game," said Leighton, who was making his second start of the season. "Sometimes you win when you're not supposed to."
The Ducks fell to 1-5 two nights after beating San Jose 4-0. Anaheim will begin a four-game road trip at Toronto on Tuesday night.
"This trip we have to use as a team-building thing," said Teemu Selanne, who has just a single goal this season. "Get the confidence there. We'll have a lot of time to hang around one another. Go from there."
"We came in here, played hard and got a road win. Their team had a burr across their rear end because they haven't gotten the start they wanted, we knew it wasn't going to be easy," said Hurricanes coach Peter Laviolette, who moved past the late Bob Johnson into second place among U.S.-born coaches with 235 career victories. Former Tampa Bay coach John Tortorella tops the list with 239.
Carolina controlled early play, taking a 1-0 lead 6 minutes after the opening faceoff when Staal sent a wraparound pass through the crease for Bayda to tap past Giguere.
The Ducks gained momentum after shutting down a lengthy 5-on-3 Carolina power play, outshooting the Hurricanes 16-5 over a 19-minute stretch before Beauchemin tied it slap shot from the right point during a delayed penalty call at 14:27 of the second.
Anaheim continued to attack after Brind'Amour broke the deadlock but could not beat Leighton to draw even again.
"When things are going against you, as they have this little while, we just haven't found a way to put the puck across the line enough times," Ducks coach Randy Carlyle said. "It's not for lack of effort or quality scoring chances. We had a territorial advantage a lot of the game."
Forward Tuomo Ruutu was scratched from the Carolina lineup because a groin injury. ... The Ducks honored several local USA Olympic medal winners before the contest. The crowd provided swimmer Jason Lezak of Irvine a lengthy ovation as a portion of his gold-medal securing anchor leg in the 400-meter relay was shown on the videoboard. ... The Hurricanes last played at Anaheim on Dec. 6, 2005.