The Bruins will face the Pittsburgh Penguins for the right to play for the Stanley Cup.
Recent call-up Torey Krug scored his fourth goal of the series for Boston, which reached the third round of the NHL playoffs for the second time since 1992. The other was 2011, when the Bruins won the sixth Stanley Cup in franchise history.
Campbell broke a 1-1 tie in the second period, then added an empty-netter with 51 seconds left after Lundqvist was pulled for an extra skater.
The Rangers fell behind 3-0 in the series before winning Game 4 on Thursday night to avoid a sweep, thanks in part to a pratfall by Rask that helped them erase a 2-0 deficit. They took an early lead Saturday, But they couldn't send the series back to New York for a sixth game.