Danny Briere scored twice and Boucher stopped 24 shots over the final two periods and overtime in a 5-4 victory on Sunday in what's been a tightly contested and topsy-turvy series that heads to the deciding game at Philadelphia on Tuesday.
Boucher took over to start the second period after Michael Leighton allowed three goals on the first seven shots he faced. Scott Hartnell had a goal and assist and James van Riemsdyk also scored for the Flyers, who overcame a two-goal deficit while drawing upon Chris Pronger's presence.
The gritty defenseman played sparingly in his game after missing 21 with a broken right hand.
Thomas Vanek scored twice for Buffalo, while Rob Niedermayer and Nathan Gerbe also scored.
Kris Versteeg helped set up Leino's goal. After getting his first shot blocked, Versteeg gained the rebound and flipped the puck at the Sabres net, bouncing it off several players. The puck fell in the crease where Leino - a step ahead of defender Chris Butler - slid it into the open side behind diving goalie Ryan Miller.
Rebounding from a pair of losses, the Flyers became the first team in the series to win after giving up the game-opening goal. It was also a game that Philadelphia didn't lead until Leino scored. Hartnell forced overtime by scoring at the 7:35 mark of a frantic third period, in which the Flyers outshot the Sabres 14-6.
Nikolay Zherdev started the play by driving up the right side and getting a shot off that Ryan Miller stopped. Mike Richards got to the loose puck and fed it across to his left to Hartnell, who shoveled it in the open side.
Philadelphia's hopes appeared done after falling behind 3-1 in the first period.
And coach Peter Laviolette had opened himself to second-guessing after going with Leighton, who became the third Flyers goalie to start a game this series. Leighton earned the start after stopping 20 shots in replacing Boucher in a 4-3 overtime loss on Friday.
It came after Boucher allowed three goals on the first 11 shots he faced.
Boucher got his second chance and his return at the start of the second period sparked the Flyers as van Riemsdyk and Briere scored 7:54 apart to tie the game. The comeback was reminiscent of the Flyers scoring three consecutive goals Friday.
This time the Sabres struck back in regulation, with Gerbe scoring with 3:51 left in the second period - and after the Flyers had killed off Buffalo's 70-second two-man advantage.
Driving across the Flyers blue line, and with four defenders back, Gerbe fired a 35-footer that hit off defenseman Sean O'Donnell's stick and beat Boucher on the glove side.
Boucher was otherwise sharp. He made a sliding save to his left to foil Drew Stafford's shot from the right circle on a 2-on-1 break early in the third period.
Miller kept the Sabres in the game in facing 45 shots - one short of matching the most he's faced in a playoff game. Pronger had a negligible impact on the ice as his playing time was limited solely to the power play.
He didn't get his first shift until 10:05 into the game and the Sabres ahead 2-0. Pronger also took an undisciplined slashing penalty while the Flyers had a two-man advantage. After being crosschecked from behind by Tyler Myers in front of Sabres net, Pronger spun around and whacked the Buffalo defenseman.
He finished with only five shifts and 4:33 of ice time.
Richards put the Flyers in trouble with 6 minutes left in the second period, when he was issued a boarding penalty for shoving Buffalo forward Tim Connolly from behind and headfirst into the boards.
Connolly, who has a history of concussion problems, lay on the ice for several minutes before getting up on his own. He was then escorted immediately up the hallway behind the Sabres bench. Connolly did not return.
NOTES: The Flyers had never had three goalies start in one playoff series. The Canucks were the last team to start three, doing so against Calgary in 2004. ... Buffalo's Lindy Ruff became the NHL's eighth person to coach his 100th playoff game with the same team. He entered the game with 57-42 record. His 100 games rank fifth among active coaches and 20th on the NHL list. ... Ruff went with veteran experience by inserting journeyman forward Matt Ellis into the lineup to replace Jason Pominville, the Sabres top two-way forward who hurt his lower left leg on Friday. ... Pominville wore a walking boot covering his lower left leg and walked on crutches.