Jason Pominville and Chris Drury each had goals during regulation for the Sabres, who claimed they were motivated all off-season by their Game 7 loss to Carolina.
Briere, the Sabres' first shootout skater, faked right and beat Cam Ward with a backhander for the only goal in the tiebreaker.
Miller made 32 saves in regulation and overtime and then came up big in the shootout. Carolina's Ray Whitney missed the net on the first attempt, before Miller stopped Eric Staal with his stick on the second, and blocked Erik Cole's weak attempt on the third.
Ward stopped 29 shots, while Staal and Cole had goals for the Hurricanes, who held one final pre-game party with the Stanley Cup before they began the gruelling task of defending their title.
During the 20-minute celebration, flashbulbs popped, those rowdy 'Caniacs whooped it up, and the Scorpions' "Rock You Like a Hurricane" blared. Then, the championship banner - commemorating the first major professional title won by a team from North Carolina - was raised to the rafters.
The Sabres took over from there, getting a win they couldn't secure in June and silencing the RBC Center crowd.
Pominville gave Buffalo an early lead midway through the first period when he redirected Dmitri Kalinin's shot from the blue line to make it 1-0. That goal came about a minute after Cole's wrist shot from close range bounced off the left post.
The Hurricanes tied it 5 minutes later when Staal took a pass from Whitney on a 2-on-1 break and flipped a knuckling shot past Miller.
Cole gave Carolina its only lead with a power-play goal midway through the second period which put the Hurricanes ahead 2-1. Drury tied it about 3 minutes later with a tip-in.