Each side had a goal during the first two shootout rounds, after which Ryan Getzlaf hit the post with his attempt for the Ducks. Terry then lifted the puck past Viktor Fasth to give the Hurricanes their second straight win.
Carolina went 4-0-1 on its five-game homestand.
It was a 1-1 game heading into the third period and Carolina took the lead on a goal by Drayson Bowman 3:01 into the stanza. Dustin Penner tied it on the power play four minutes later.
Anaheim suffered its second overtime loss of the season and has lost the opening three outings in a four-game road trip.
The win went to Justin Peters, who made 28 saves. Fasth stopped 21-of-24 shots.