Campbell's tally off a loose puck in the crease with 6:19 left in the second period broke a 1-1 tie and put the Bruins on the path to their series-clinching fourth victory in five games over the Rangers.
Campbell added an empty-net insurance goal and Torey Krug also scored as fourth-seeded Boston reached the conference finals for the second time in three seasons.
Tuukka Rask turned aside 28-of-29 shots, including all 12 he faced in the final period.
Boston now advances to face the Pittsburgh Penguins, who defeated Ottawa 4-games-to-1 in the other East semifinal.
Dan Girardi scored and Henrik Lundqvist made 29 saves for sixth-seeded New York.