David Krejci had two goals and one assist, and Rask made 27 saves on Monday night as Boston beat Buffalo 4-3 in Game 6 and advanced to the Eastern Conference semifinals.
The Bruins will play Philadelphia if Montreal beats Washington in Game 7 on Wednesday night, or Pittsburgh if the Capitals beat the Canadiens in the only Eastern Conference series still in doubt.
"We wanted to close it out because we didn't want to go back to Buffalo," said Rask, who allowed an average of just over 2 goals per game. "We knew it would be tough."
And they advanced with help from a goalie who won the starting job away from U.S. Olympic backup Tim Thomas in the regular season, then knocked out the American starter in the first round of the postseason.
"As good as he was, he was arguably facing the best goaltender in the league, if not more," Bruins coach Claude Julien said. "He obviously made a reputation for himself. For him to go head-to-head and come out the winner, I think it speaks volumes for how Tuukka's played."
Miller made 28 saves for Buffalo. But he was off the ice for an extra skater for much of the last 2:25, when the Sabres managed to cut a two-goal deficit to 4-3 when Thomas Vanek scored with 1:13 left.
The Sabres couldn't get any closer.
"It was an extremely tight series. The teams couldn't separate. There wasn't much breathing space the whole series," coach Lindy Ruff said. "They scored two power-play goals, and we didn't get any. We made some mistakes in the series, some mistakes we'd like to have back. Overall, it wasn't good enough. That's the bottom line."
The Sabres finished the regular season with 100 points, a Northeast Division title and a third seed that gave them home-ice advantage over the sixth-seeded Bruins. Buffalo scored first in every game except the last, but the Sabres' power play went 0 for 19 in the series, failing in three more tries on Monday.
"It's too late to look at it now," forward Jason Pominville said. "It is probably the difference between winning and losing the series."
The Sabres pulled Miller trailing by two goals, but had trouble keeping him off the ice. After a faceoff at center ice with just over two minutes remaining, Vanek, who missed the previous three games with a sore left ankle, put them within one goal.
After killing off the final minute, the Bruins had a muted celebration on the ice while the fans threw the yellow towels they had been given over the glass.
The Bruins led for just 19:40 of the first 327:41 in the series. But Boston took the lead on Monday night when Recchi's pass was redirected behind Miller by Krejci at the 13:39 mark of the first period.
The two switched roles just 61 seconds into the second period, when Boston had a 4-on-3 advantage and set up some quick passing from Krejci to Recchi on Miller's left. Boston took a 3-1 lead with 12:42 left in the game when Satan dug out a puck on the side and pushed it behind the net, where Lucic backhanded it in front to Krejci.
But just 22 seconds later the puck came free into the middle to Gerbe, a former Boston College star, who one-timed it into the net to make it a one-goal game again. With 5 minutes left, Satan made a move to get Miller out of position but hit the post. Then just a few seconds later, he waited for a pass from Dennis Wideman at the side of the net and tucked it around Miller to make it 4-2.
The Bruins hadn't clinched a playoff series at home since May 2, 1999. ... Vanek was the Sabres' leading scorer in the regular season, with 25 goals and 28 assists. ... It's the first time Boston has won playoff series in consecutive years since 1991-92. ... Recchi recorded his 75th career playoff assist. ... Kaleta and Gerbe each scored his first career playoff goal.