Game 7 is Monday night in Boston.
Not only did the win keep the Leafs alive, it snapped a 54-year run of home playoff failures against the Bruins.
Toronto's previous home playoff win against Boston was March 31, 1959, when the Leafs won 3-2 in overtime. Nine straight postseason home losses followed in the decades since.
Milan Lucic scored for Boston with 26 seconds left in the third.
The only time Toronto has come back from a 3-1 deficit to win a series was the 1942 Stanley Cup finals against the Detroit Red Wings. The Leafs trailed 3-0 in that series before reeling off four straight wins.
Toronto's last Game 7 win was in 2004 the last time the Leafs made the playoffs when it beat Ottawa 4-1 to wrap up a first-round series.
The Maple Leafs have momentum after two consecutive wins.
James Reimer made 29 saves for the Maple Leafs.
For Phaneuf, scoring a goal was a bit of redemption after being involved in the play that led to the Bruins' overtime winner in Game 4.
Phaneuf's goal at 1:48 of the third period came after Nazem Kadri ripped a wrist shot that was tipped in by Phaneuf, who had made his way to the front of the goal after continuing his rush.
Boston had lost the puck in the Toronto end on an attempt at a pass by David Krejci.
Kessel then added to the lead at 8:59, picking up the puck after James van Riemsdyk dealt with two Bruins in front of goal and backhanded it past Tuukka Rask. Kadri beat Patrice Bergeron on the faceoff to start the play.
Rask finished with 24 saves.
After combining for 170 shots in the past two games, the teams played defensively in what started as a much tighter game, but opened up later on.
Reimer and Rask were unbeatable in the first two periods, adding to the importance of the first goal.
The tight game made for a tense atmosphere inside where 19,591 fans, wielding giveaway blue or white towels.
Outside, another amped-up crowd packed Maple Leafs Square outside the Air Canada Centre to watch the game on a big screen, despite chilly temperatures at game time.
It was another strong game for Reimer, who switched masks during the game.
Rask was equally impressive, showing off a lively glove.
First-line Leafs center Tyler Bozak skated during warm-ups but was scratched.
Bozak, who scored short-handed in Game 5, had missed the final two games of the regular season with what was believed to be a shoulder injury.
Joe Colborne, making his playoff debut, came in for Bozak.
Boston defenseman Andrew Ference did not make the trip to Toronto for undisclosed reasons. But Wade Redden returned to the lineup after missing Game 5.
That prompted Bruins coach Claude Julien to change his defensive pairings, as he did when Ference was suspended for Game 2. Captain Zdeno Chara played with Johnny Boychuk, Dennis Seidenberg with Adam McQuaid, and Redden with rookie Dougie Hamilton.
Boston dominated most of the first period, skating hard and pushing the pace.
Toronto had to kill off an early penalty. Leafs coach Randy Carlyle juggled his forward lines.
Shots were few and far between but Boston's Gregory Campbell, parked in front of the net had a good chance near the midway point of the period. But he failed to get a lot on the attempt and Reimer made the stop.
The Bruins outshot Toronto 8-7 in a tight first period. Without Bozak, the Leafs won just six of 17 faceoffs in the period. Toronto outhit Boston 19-17.
Boston continued to bottle up the Leafs in their own end for stretches of the second, with Boychuk hitting the post with a shot from the blue line.
A diving Reimer made a stop on Patrice Bergeron six minutes in, going across the crease on his stomach as Boychuk came from behind the net and tried to stuff the puck in.
Reimer, who came into the game leading all playoff goalies in shots against (207) and saves (192), stopped Bergeron again seconds later.
After the Game 5 loss, Julien called for more production from the line of Bergeron, Brad Marchand and Tyler Seguin, who had 59 shots but just one goal in the first five games.
The trio combined for one shot in the first period, but were more active as the game wore on.
Toronto swarmed the Boston goal midway through the second period, with Jake Gardiner pressuring the Bruins, and Kessel forced a nifty glove save from Rask.
The line of Krejci, Lucic and Nathan Horton, which came into the game with eight goals and 15 assists, was again dangerous for Boston and Krejci came close in the second, but fired wide on an odd-man rush.
Rask showed off his glove skills late in the period when he caught a Phaneuf shot.
The teams had 10 shots apiece in the second period. Toronto led the hit count 42-33 with Leafs defenseman Ryan O'Byrne and forward Leo Komarov leading the way with six each.
The Leafs came into the game 1-13 in best-of-7 series in which they trailed 3-1 and 6-14 in series in which they were behind 3-2.