Jason Pominville had two goals and two assists, and Thomas Vanek had a goal and an assist as the Sabres came back from a 4-2 deficit in a game that featured three lead changes. Stafford capped his fourth career hat trick by forcing overtime with 28 seconds left.
Zdeno Chara and Tyler Seguin had a goal and assist each for the Bruins, 3-0-2 in their past five. Steven Kampfer, Dennis Seidenberg, Brad Marchand and Andrew Ference also scored for Boston.
After Ennis gave the Sabres a 3-2 lead in the shootout, Chara lost control of the puck and shot wide on the Bruins' final chance.
Tim Thomas stopped 16 shots over the final two periods and overtime in place of Tuukka Rask, who was pulled after allowing three goals on 16 shots in the first period. Thomas didn't stop a shot during the shootout, allowing Pominville, Vanek and Ennis to score.
Ennis drove to the net and got Thomas completely turned around before flipping the puck just inside the left post.
The Bruins squandered a 4-2 first-period lead, and dropped to 19-1-3 in games in which they led. Two of those losses were to Buffalo. Boston is 19-1-2 after scoring the first goal.
The Sabres couldn't hold a 5-4 lead after Pominville scored a power-play goal with 2:17 left in the second period.
The Bruins appeared to take control when Seguin and Kampfer scored 3:58 apart to put Boston ahead 6-5 7:06 into the third.
Buffalo, coming off a 4-2 win at Edmonton on Tuesday, won consecutive games for the first time since Dec. 3-4.
The game featured a wide-open first period in which the Bruins took a 3-1 lead 5:36 in when Marchand tipped Johnny Boychuk's point shot past Miller.
Stafford cut the Bruins' lead to 4-3 with 40 seconds left in the first period on a weak goal. Stafford caught Rask cheating off the right post and attempted to sneak in a shot from a bad angle. Rask got a piece of the shot, and the puck fell at the post, where the goalie attempted to cover as Stafford got his stick in and shoved it over the line.
Thomas took over in goal to start the second period.
The combined seven goals in the first period marked the most in a Sabres game since a 7-4 loss to Boston on Nov. 19, in which Buffalo led 4-3 after 20 minutes.
Notes
The Sabres inducted forward Alexander Mogilny and Hall of Fame hockey writer Jim Kelley, who died last month, into the team's Hall of Fame. Mogilny had 211 goals and 444 points in 381 games in Buffalo. His 76 goals in 1992-93 remain a franchise single-season record. Kelley covered the Sabres for more than 20 years with The Buffalo News. … Sabres C Tim Connolly returned after missing four games with a broken nose. … Bruins C Gregory Campbell missed his second straight game due to an illness. … The Bruins will complete a season-high, five-game road trip at Toronto on Monday.