Phaneuf netted his first career playoff goal on the redirection of a Nazem Kadri shot from the high slot 1:48 into he final frame, breaking a scoreless tie.
Kessel doubled the Maple Leafs' advantage at the 9-minute mark, backhanding a loose puck in the crease over Boston goalie Tuukka Rask for what turned out to be the game-winner.
James van Riemsdyk assisted on both tallies as Toronto won in a do-or-die Eastern Conference quarterfinal contest, setting up a deciding seventh game Monday in Boston.
James Reimer turned aside 29-of-30 shots in backstopping the Leafs to their first home playoff victory since 2004.
Milan Lucic got one back for the Bruins with 26 seconds remaining, but Boston could get no closer.
Rask finished with 24 saves.