Grier netted the winner with 3:32 remaining when, in traffic, he tipped Chris Drury's shot while standing in front of the net.
"We just got it in deep and Chris turned and shot," Grier said. "I was just fortunate enough to turn around and tip it."
It was the second deflected goal for Buffalo. Jay McKee's shot went off Grier's knee and off Drury's stick before it got past goalie Dwayne Roloson with 39 seconds left in the second period, tying the game at two.
The goals and game were typical of the way the Sabres have played during their 11-1-1 run, which included seven consecutive road wins. Buffalo's run started right after a loss at Ottawa on Nov. 12.
"When games are close in the third period or tied we have the confidence that we can come back and win that game or tie it up," Grier said.
Biron made 34 saves to post the best winning streak by a Sabres goalie, breaking the mark of nine set by Gerry Desjardins in the 1976-77 season.
The last NHL goalie to win 10 consecutive starts was Detroit's Dominik Hasek in 2001-02.
"It's something to build on. You don't want to stop at 10," Biron said. "You feel that things are working pretty good and you want to keep going in that same direction."
"When we've needed him to close the door, he's done it for us," Buffalo coach Lindy Ruff said.
Brian Rolston had a goal and an assist for Minnesota, which has lost six of seven. The Wild returned home following a 1-3-1 road trip in which they scored just 11 goals. They are 0-12-4 when scoring two goals or fewer.
"We need to be desperate and find a way to score," defenseman Willie Mitchell said. "We had chances, but we weren't converting."
Minnesota, which allowed a goal in the final minute of a 3-2 loss Saturday at Philadelphia, tied it three minutes later on a power play. Mikko Koivu took a perfect feed from Robitaille and beat Biron with a one-timer from the slot.
Rolston tipped a pass from Marian Gaborik past Biron at 11:47 of the second period to put Minnesota up 2-1.
Gaborik had an opportunity four minutes later when he took a pass from Pierre-Marc Bouchard and got a stride behind two Sabres defenders, but his shot hit Biron in the chest.
Wild coach Jacques Lemaire said his players should have no problem continuing to work hard, even though they haven't been rewarded recently.
"First of all, it's their job," he said. "They get down after the game. When it happens, you live with it for a little while, and after they know it's a new day, it's a new game, so you got to scratch, you got to skate and you got to keep going."
Buffalo, which entered the game with the NHL's second-best power play at 21.7 percent, went 0-for-5. Minnesota was 1-for-8.
- The game was the 200th consecutive sellout for Minnesota, including preseason and playoff games. The Wild have sold out every game in team history.
- Minnesota D Filip Kuba missed his third straight game with back spasms. LW Pascal Dupuis missed his second straight due to the flu.
- C Daniel Briere returned for Buffalo after missing eight games with an abdominal strain. He had four shots on goal.
- Vanek scored 113 points in two seasons at the University of Minnesota.