Briere reached the milestone when he banged in a rebound from the slot with 0.4 seconds left in regulation and the Avalanche beat the Boston Bruins 2-1 for their first win of the season.
Colorado hemmed the Bruins in their own end for most of the final 30 seconds of regulation before Jan Hejda landed a shot on Bruins goaltender Niklas Svedberg and the puck deflected to a wide-open Briere in the slot.
"Hopefully there are many more to come but it's certainly a good feeling to find a loose puck like that late in the game and get a big win, especially after the start that we had," Briere said. "It's been a little bit of a roller coaster to start the season but we are hoping this game will get us on our way."
The Bruins have lost three in a row in regulation for the first time since losing four straight March 10-15, 2012.
"I think overall I think our team played better," Bruins coach Claude Julien said. "I think it's a bitter pill to swallow right now because you lose a game in that fashion. But had it been the other way, had we won the 2-1 game, I think everybody would have been fairly pleased with the effort.
"I think it was a better effort. I think there was better concentration. I think at the end of the day, some of those crucial mistakes came back to bite us. That's the story of this afternoon's game and it's unfortunate. Like I said, it's a bitter pill to swallow. But I saw some improvement in our team overall and I saw some positive things happen. Now we've just got to turn these things into wins."
The Avalanche were shut out in their first two games by the Minnesota Wild and didn't snap their drought until McGinn's goal in the second period ended the drought at 143:27 from the start of the season.
In the second period, the Avalanche took advantage of a Bruins icing by getting the line of Matt Duchene, Jarome Iginla and McGinn on the ice for an offensive-zone faceoff. Off the draw the Avalanche controlled play and Tyson Barrie's pinch down the right wall led to McGinn skating toward the net with the puck on his stick. The left wing took a wrist shot that beat Svedberg under his left arm for a 1-0 lead at 3:28 of the second period. The goal allowed the Avalanche to avoid a measure of infamy; according to the Elias Sports Bureau, the only team in NHL history to be shut out in its first three games of the season was the 1930 Montreal Maroons.
The Bruins had to solve their own scoring woes. They were shut out by the Washington Capitals on Saturday, and Loui Eriksson's power-play goal in the second period tied the game 1-1 and ended their drought at 135:48 dating back to a 2-1 loss to the Detroit Red Wings last Thursday.
After center Carl Soderberg knocked down teammate Reilly Smith's wrist shot, he settled the puck and found Eriksson cutting across the slot. Eriksson waited with the puck on his backhand and then flipped it over a sprawled-out Berra.
Svedberg stopped 28 of 30 shots in his first start of the season. Berra made one relief appearance in the Avalanche's first two games, but Monday was his first start.
"He had a great training camp and he just continued to play well," Avalanche coach Patrick Roy said. "It was nice to see him make some good saves for us in the first period. I think they had five or six good scoring chances in the first and he made those saves which gave us time to take the lead, and it was nice for us to play with the lead. They came back on that power play but I thought our guys were focused, I thought our guys were skating well and competing well, it was a real nice win for us."
Colorado defenseman Ryan Wilson played six shifts for a total ice time of 6:07.
"He got a puck in the face and I think he had a concussion," Roy said. "He's probably not going to play [Tuesday]."
The Avalanche have won seven straight at TD Garden and are 8-0-1 in their past nine games in Boston. The Bruins last defeated Colorado on home ice on March 30, 1998.
Colorado and Boston will complete their two-game season series in Denver on Jan. 21, 2015.
The Avalanche play the second game of their four-game road trip Tuesday at the Toronto Maple Leafs. The Bruins next play Wednesday on the road against the Detroit Red Wings.