The win, Buffalo's sixth straight at home against Boston, snapped the Sabres' season-high, three-game regulation-losing streak.
"It's a step in the right direction," said Briere, who played his first home game since earning All-Star MVP honors last Wednesday in Dallas. "We don't want to get too confident because it's just one game, but we needed this to realize that we're OK here."
Briere has been better than that lately. The co-captain is on a six-game point streak (5 goals, 6 assists), and had five points in the All-Star game.
"He's a heck of a player," Pominville said. "He's still proving what he is capable of."
The Sabres squandered third-period leads in consecutive losses to Columbus and the New York Islanders last weekend.
"We were struggling with the one-goal leads, so tonight we made sure we didn't stop," Briere said. "We just kept scoring goals."
Chris Drury and Paul Gaustad also scored, and Maxim Afinogenov posted three assists for Buffalo, which went 6-7-1 in January. It was the Sabres' worst month since December 2003.
"We did the little things better, and that's what made the difference," Buffalo coach Lindy Ruff said. "There was desperation in a lot of areas."
The Sabres scored two power-play goals, including Vanek's 25th tally just 2:15 in while Buffalo had a two-man advantage. They also had a short-handed goal when Pominville wristed a shot past goalie Hannu Toivonen at 15:11 of the second, pushing the lead to 5-1.
Buffalo broke the game open with four goals in the second, including two by Briere. Briere and Gaustad scored 79 seconds apart 6 minutes in, and Briere capped the big period with a goal at 17:24.
He scored his third with 8 minutes left in the game, lifting the puck over sprawling goalie Tim Thomas, and became the fifth Buffalo player to record 20 goals this season. The Sabres are the only NHL team with five 20-goal scorers.
Thomas replaced Toivonen to start the third period and finished with 11 saves. Toivonen stopped 17 shots in his third start in five games.
Glen Murray scored for the Bruins, who lost their seventh straight road game. Boston has been outscored 19-3 during a four-game losing streak overall.
"It's a struggle," Bruins coach Dave Lewis said. "It's a situation nobody likes to go through, but hopefully we'll come out stronger."
Buffalo backup Martin Biron, inserted into the nets to give Ryan Miller a break, made 25 saves in winning for the first time since Jan. 6.
Murray cut Boston's deficit to 2-1 with his team-leading 24th. He scored while the Bruins had a two-man advantage. The Sabres then scored five unanswered goals to secure their third win in nine games.
"We've been struggling lately," Pominville said. "It's only one game, but I think we're back on track."
Boston, meanwhile, has been outscored 34-9 during its road skid.
"So many things are not going our way right now," defenseman Zdeno Chara said. "We need that spark, that break. Sometimes that's all you need."
Notes: Toivonen relieved Thomas on Monday during the second period of Boston's 6-1 home loss to the New York Rangers. ... Sabres D Henrik Tallinder returned from an ankle injury that forced him to miss 11 games. ... The teams will meet again Thursday in Boston, their fourth matchup in 18 days. ... Boston C David Krejci made his NHL debut after being recalled earlier in the day from Providence of the AHL. He was knocked out of the game with a concussion 5 minutes left in the first period after a hit by Sabres forward Adam Mair. ... Bruins D Andrew Alberts was hit in the face by Brian Campbell's slap shot late in the first and left the ice bloodied. He returned for the second period.