Jakub Zboril scored his first NHL goal for Boston, and Brad Marchand had two assists. The Bruins won for the 10th time in 11 games.
It was Kinkaid's first game with Boston. He made 20 of his saves in the first two periods.
''They came out early and often, so it was good to get into the game right away and see some far shots,'' Kinkaid said. ''We prevailed in the end, not the best first two periods but I just wanted to give my team a shot and they did the rest.''
Kinkaid, who signed with the Bruins in July as a free agent, replaced backup Jeremy Swayman, who is week to week with an undisclosed injury.
''You look at what we're doing on the ice and it's incredible depth,'' Bruins coach Jim Montgomery said. ''You've got to give credit to our pro scouts who went and signed Keith Kinkaid and the depth we have through our organization.''
Bergeron closed it out when he scored with 1:49 remaining in the third period. It was his seventh goal of the season.
Tage Thompson scored his 11th goal in Buffalo's fifth consecutive loss. Craig Anderson made 27 saves.
''We've got a group that - I've said it before - they're not happy,'' Sabres coach Don Granato said.
''That's a really good thing for us, is we brought in competitive people. ... When this stuff happens, they don't like it. They want to fix it.''
Thompson put the Sabres in front with a short-handed goal 5:17 into the first, extending his point streak to four games.
''Tage is a one-man wrecking crew out there,'' Anderson said. ''When the puck's on his stick, you know something explosive is going to happen.''
Boston's league-leading penalty kill stopped all three Sabres power plays, including two in the second period after back-to-back penalties on defenseman Charlie McAvoy.
''It wasn't a Picasso moment out there in the first two periods besides the special teams and our goaltender,'' Montgomery said.
Bergeron tied it at 1 with 1:09 left in the second. The captain scored a power-play goal from his knees when he put a rebound of a Jake DeBrusk shot past Anderson.
Zboril put Boston ahead to stay with 7:29 left in the third with a screened shot from the blue line.
Sabres captain Kyle Okposo was sidelined by a lower-body injury. He missed practice on Friday and did not participate during Saturday's morning skate. He is listed as day to day. He was replaced in the lineup by rookie Jack Quinn.
ZBORIL MAKES HISTORY
Jakub Zboril became the first defenseman in Bruins history to have his first career goal stand as the game-winner.
Bruins: Host Vancouver on Sunday in the first game of a three-game homestand.
Sabres: Host Vancouver on Tuesday to conclude a four-game homestand.
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