The Bruins were trailing 1-0 three minutes into the final frame when Boychuk's snap shot from above the right circle found its way through traffic and into the net for the equalizer.
Then with 3 1/2 minutes left in regulation, Rangers goalie Hendrik Lundqvist lost sight of the puck after stopping a Shawn Thornton shot and Paille batted in the loose disc for the decisive tally.
Tuukka Rask turned aside 23-of-24 shots for Boston, which will seek to sweep the series in Thursday's Game 4 at Madison Square Garden.
Taylor Pyatt a scored and Lundqvist finished with 32 saves for the Rangers.
2014: The Year in Music [PHOTOS]