Boston trailed twice at TD Garden, to goals from Devante Smith-Pelly and Mathieu Perreault, but levelled through first Carl Soderberg and then a late Zdeno Chara strike.
Jarome Iginla netted the only goal of the shoot-out to take the Bruins second in the Atlantic Division, while Anaheim remain in the equivalent spot in the Pacific Division.
The Phoenix Coyotes are only a point behind the Ducks after a shoot-out win of their own, 5-4 over the Nashville Predators.
Nashville led 3-0 after the first period and 4-2 early in the third, but Shane Doan and Antoine Vermette forced overtime and when there was no further score, Mikkel Boedker finally settled the match with Phoenix's fifth effort of the shoot-out.
The New York Rangers beat the Buffalo Sabres 2-0 in the night's other game. Henrik Lundqvist had a 29-game shut-out and Derick Brassard and Chris Kreider scored the goals.