Anaheim caught Carolina on a change, and Strome finished a 2-on-1 with Cam Fowler for his second overtime goal this season.
"I wanted to pass, but the shot was there and I shot it," Strome said. "Lucky it went in. Nice play by Cam and I thought we possessed the puck well in overtime."
John Gibson made 34 saves, and Adam Henrique, Max Comtois and Brett Leason scored for the Ducks (7-17-3), who went 0-4-2 during their slide.
"They're a quick team," Anaheim coach Dallas Eakins said. "They're one of the elite teams in the League, and to get two points against a team like that is certainly encouraging."
Brady Skjei, Sebastian Aho and Jordan Staal scored, and Pyotr Kochetkov made 26 saves for Carolina (14-6-6), which had its four-game winning streak end but has an eight-game point streak on the road (4-0-4).
"Mentally, you could tell, right from the start of the game, we were not ready to go," Hurricanes coach Rod Brind'Amour said. "Not from a lack of effort, just the sharpness was not there."