Jansen Harkins, Radko Gudas and Ryan Strome scored, Mason McTavish had two assists, and Lukas Dostal made 36 saves for the Ducks (13-6-1), who have won six straight at home.
Morgan Geekie scored two goals, Michael Eyssimont also scored, Hampus Lindholm had two assists, and Joonas Korpisalo made 29 saves for the Bruins (12-10-0), who have lost three of four following a seven-game winning streak.
Moore scored with a one-timer from the high slot to move the Ducks back ahead 4-3 after they surrendered a two-goal lead.
Anaheim scored on its first shot of the game to take a 1-0 lead at 2:29 of the first period.
Ross Johnston made a centering pass from just above the goal line that went off Korpisalo's stick and continued to Harkins, who pushed the puck into the net as he was closely defended by Lindholm.
Gudas scored his first goal of the season on a slap shot from just inside the blue line that went off the skate of Boston forward Fraser Minten and into the net for a 2-0 lead at 6:29.
Geekie was credited with Boston's first goal when he deflected a point shot from in front of the crease while on a power play. The puck got behind Dostal and Anaheim defenseman Jackson LaCombe tried to save it, but knocked the puck across the goal line to cut it to 2-1 at 14:58.
After the Ducks had a goal disallowed following a collision with Korpisalo, Strome scored his first of the season on a deflection with two seconds left on a power play to extend the lead to 3-1 at 13:47 of the second period.
The Bruins cut it to 3-2 at 18:27 of the second when Eyssimont pulled up in the right circle and scored short side with a wrist shot.
Geekie scored on another deflection seven seconds into a power play to tie it 3-3 at 7:39 of the third period.
