Defenseman Jay McKee scored 18:02 into overtime after Hasek made a series of spectacular saves to give the Sabres a 4-3 victory and a commanding two games to none lead in their Eastern Conference quarterfinal series.
Buffalo scored just two goals in losing all four regular-season meetings with Philadelphia. But the Sabres won the first two games on enemy ice and return home for Game Three on Monday.
"We feel very good as to where we are. You can't get any better being up, 2-0," Hasek said. "We know that they've won in Buffalo twice, so we have to be really sharp because they have proven that they can win in our building. Nothing is over, we know that, and it's possible that we will back in this building next week."
Philadelphia had the better of the scoring chances in overtime before Erik Rasmussen slid a cross-ice pass to McKee, who trailed the play before putting a wrist shot from the left faceoff circle along the ice and under the glove of Flyers goaltender Roman Cechmanek.
"I hopped on the ice after a line change and I went to the Flyers' zone. I saw Rasmussen turn up the ice. I saw a small opening and went there" said McKee, who had just one goal in 74 regular-season games. "I got a great pass from Erik, took one look and I just tried to get the shot low on the net and hope for the rebound. I was lucky it caught the far side of the net. I don't get many goals, so this definitely is the biggest."
Hasek demonstrated his brilliance earlier in overtime, thwarting a series of good scoring chances. Dean McAmmond was stopped on a breakaway with 12 minutes left and Hasek made a diving save on rookie Ruslan Fedotenko 5 1/2 minutes later.
The best chance for Philadelphia might have been earlier in overtime, when a deflection off Sabres center Stu Barnes nearly got past Hasek.
"We had a lot of good scoring opportunities, but Hasek made several big saves for them," Flyers left wing John LeClair said. "As long as we keep getting good opportunities, we'll get some goals. We felt we were going to pull through. But it just didn't happen."
While Hasek, who made 32 saves, has stepped up in the playoffs, Cechmanek has allowed some soft goals in the first two games, including McKee's game-winner. Hasek's backup on the 1998 Czech Olympic team, Cechmanek shut out the Sabres twice during the season. He stopped 37 shots on Saturday.
Buffalo's Curtis Brown scored the only goal of the third period to tie it at 3-3. He won a faceoff and the puck slid toward to the net, where it went off Vaclav Varada's stick. Brown swooped in and beat Cechmanek to the stick side.
Miroslav Satan had a goal and an assist and defenseman Alexei Zhitnik scored a fluke goal for the Sabres.
"I never would have predicted anything like this," Buffalo coach Lindy Ruff said. "We won by the narrowest of margins."
The Flyers controlled play for much of the first period and scored the only goal at 13:13 during a two-man advantage. LeClair held the puck for several seconds at the bottom of the right circle before feeding a pass through the crease to Daymond Langkow, who tapped it past Hasek.
Philadelphia had a 10-4 edge in shots in the opening period and the Sabres did not test Cechmanek until 4:16 remained.
But Buffalo tied it on the power play 47 seconds into the second period. Former Flyer Chris Gratton, skating along the goal line, sent a pass through defenseman Chris Therien's legs to a streaking Satan, who directed the puck past Cechmanek.
Buffalo moved ahead 2-1 on a fluke goal credited to Zhitnik at 9:40 of the second. Barnes, who had lost his stick, kicked the puck back to the blue line to Zhitnik, whose short slapper went off defenseman Dan McGillis' right skate and through Cechmanek's pads.
With the game seemingly slipping away, LeClair tied it with 5:29 left in the second with a slap shot from the left circle that beat Hasek to the short side.
Mark Recchi gave the Flyers their only lead less than two minutes later, poking a puck through Hasek's pads off a scramble in front.
With both games so close, Recchi is not among those conceding defeat. In fact, he guaranteed a win.
"It's frustrating, but we're not going to quit," Recchi said. "We'll go up there and win on Monday, get to 2-1 and we'll go from there. We'll be there to fight."
The game was delayed for nine minutes midway through the first period when a pane of glass along the boards was shattered by a shot.