Briere scored a spectacular power-play goal 79 seconds into the third, going end-to-end, cutting around defenseman Marcus Ragnarsson and beating goaltender Evgeni Nabokov to the glove side from the right faceoff circle to give the Coyotes a 2-0 lead.
"I got a little bit lucky, going behind the two defensemen," Briere said. "I think they were caught standing still. When I came into Nabokov, there wasn't much room. He covered pretty much everything and I couldn't even see it when I shot it. It found a hole."
"The second goal was the key to the game because it makes it a 2-0 game," Nabokov said. "It is always really tough to come back down two goals. It was a good shot. I went down and it went right over my glove."
Phoenix goaltender Sean Burke recorded 27 saves, recovering from a blunder that cost the Coyotes the first game to win only his third playoff game since 1988.
"I had a couple of tough games against these guys earlier in the year," Burke said. "I just figured this series has tightened up a little bit more. They had a couple good chances in the second period, but I was able to make those saves."
In the series opener, Burke strayed too far out of the net, leading to Patrick Marleau's game-winner. The loss dropped Burke to 2-10 in the playoffs since his rookie season with the New Jersey Devils.
But he was sharp throughout Game 2 and came within 5:41 of only his second career postseason shutout.
"That is a team over there that is a very gritty team and tough to play against," Burke said. "We have to play even better and get better every game. Every game is going to be a battle. We have to get the next one."
Marco Sturm was left alone in front and got the Sharks on the board when he took a cross-ice feed from Marleau and tapped the puck into an open net.
But Doan answered 40 seconds later as the Sharks got caught on a two on one. Nabokov kicked out Denis Pederson's slap shot, but Doan banged in the rebound to restore Phoenix's two-goal cushion.
"I don't know how the puck ended up in a two on one, but obviously they were pressing, being down a goal, to create things" Doan said. "The rebound came out to me and it was nice."
Phoenix also got a boost from the return of top defenseman Teppo Numminen, who missed Game 1 with a broken left foot.
By splitting the first two games on the round, the Coyotes grabbed home-ice advantage in the best-of-seven series and will host the next two on Monday and Wednesday nights. They have not gotten past the first round of the playoffs since 1987, when the franchise was based in Winnipeg. Since then, the Coyotes have dropped 10 consecutive series.
"I don't think the home-ice advantage matters very much," Briere said. "Playing against the Sharks, you don't really have a home-ice advantage, but it will be fun going back to our fans. It's going to be a long series."
San Jose, which opened a series at home for the first time in team history, missed a chance to go up two games to none for the first time since 1995, when they went up 2-0 in a seven-game Western Conference quarterfinal victory over Calgary.
After a tight-checking first period, Daymond Langkow gave the Coyotes a 1-0 lead when he poked in Doan's rebound 5:31 into the second. San Jose pressed for the equalizer, but Burke turned aside all 14 shots he faced in the period.
"It is going to be tough going to Phoenix," Nabokov said. "They are a very young team and they don't have anything to lose. We need to be strong all 60 minutes, put the puck in the net and play better on defense."