Miller was given a five-minute major and game misconduct for knocking Burmistrov out of the game in the second period. The Coyotes center remained motionless on the ice for several minutes but gave a thumbs-up with both hands as he was wheeled off on a stretcher.
The Coyotes said he was taken to a hospital for further evaluation.
Boston has won six of seven games since firing coach Claude Julien and replacing him with Bruce Cassidy.
Tuukka Rask stopped 22 shots for the Bruins. Mike Smith stopped just 10 of Boston’s 14 shots in the first two periods. He was replaced at the start of the third by Louis Domingue, who had 11 saves.