The victory was Phoenix's first in 11 games in Anaheim dating to Nov. 12, 2003.
"When you come into this building, you expect a lot of work," said Auld, who made many of his 31 saves with heavy traffic in front of him. "They are a very good team. We really weathered the storm.
"I thought we showed great poise for such a young team. That's huge for us going forward."
Phoenix coach Wayne Gretzky said Auld was solid, and: "We got one lucky break, one lucky goal and we won the hockey game. I can tell you a couple of games we thought we should have won that we didn't. But it kind of evens out by the end of the year."
Auld signed as a free agent in August and the rangy 6-foot-5, 221-pounder evened his record with the Coyotes at 2-2.
Weller, who also signed as a free agent over the summer, scored at 3:02 of the second period.
"There was a lot of determination on our part and our 'D' played well," Weller said. "We didn't panic out there. They had some shifts where they had a lot of energy and they were throwing a lot at us.
"But we stayed together, we stayed in the middle of the ice and kept them to the outside."
Weller, a sixth-round pick by St. Louis in the 2000 entry draft, played at Hartford of the AHL for the past four seasons and was the team captain last year.
The defending Stanley Cup champion Ducks are off to a sluggish 4-7-1 start.
"I think the biggest thing is for us not to panic and start pointing fingers," Chris Pronger said. "We're all making mistakes out there. It's a fine line between winning and losing. Those little mistakes we're making are costing us games.
"It can be frustrating at times, but those are the situations when you don't want to be frustrated. You need to take a step back, take a couple deep breaths and make sure that you're staying as focused and determined as possible."
Bryzgalov had 20 saves, but still lost to the Coyotes for just the second time in his nine games against them.
The Ducks went 0-for-6 on power plays, including one 15-second span of the first period when they had a two-man advantage. The Coyotes had four power plays.
- C Ryan Getzlaf, Anaheim's leading scorer with 10 points, was sidelined by an upper body injury and F Rob Niedermayer was out with a groin injury. Also missing for the Ducks were D Mathieu Schneider (ankle) and F Todd Bertuzzi (concussion).
- Auld's most recent shutout had been Nov. 28, 2006 against Montreal.
- The Ducks have been busy so far this season, which they opened in London on Sept. 29-30. They've played 12 games so far, half of them on the road, and they still haven't won away from their home ice.