"We were creating a lot of opportunities," Carolina's Eric Staal said. "We knew that we'd eventually break him."
Semin tied it at 2 on a power play with 8:12 left, and Tlusty scored the game-winner with 6:17 to go. Carolina put up a season-high 55 shots against Neuvirth.
"The guys did a good job of not getting frustrated," Hurricanes coach Kirk Muller said. "We talked after the second period and said, `Let's take a breather, get our rest and let's focus -- and let's play the same way."
Neuvirth's season-high 51 saves came in his second start for the Sabres since being acquired in a trade with Washington on March 5.
"I thought he was good again," Buffalo coach Ted Nolan said. "The one thing our goalies are going to end up doing in the remaining 16 games we have -- they're going to get a lot of good looks and play for that position."
Tlusty's winner came after he took a pass from Semin, walked into the slot and beat Neuvirth with a wrist shot above the glove.
Stafford scored with 11:17 left to give Buffalo a short-lived 2-1 lead.
The Sabres' Cory Conacher attempted a wraparound on the play, but the puck bounced off the skates of Tlusty and Carolina defenseman Andrej Sekera in the crease before coming to Stafford at the side of the net. Stafford snapped it past Ward for his third goal in three games. He has nine goals in his last 15 contests.
Carolina tied it when Semin's slap shot from the wing deflected off the skate of Buffalo defenseman Mike Weber in front of Neuvirth. It was only Carolina's second power-play goal since the Olympic break. The Hurricanes came into the game mired in a 1-for-32 slump with the man advantage.
Buffalo opened the scoring midway through the opening period after Tlusty was called for slashing Ennis on a breakaway. Official Eric Furlatt called a penalty shot for the infraction, and Ennis beat Ward with a rising shot above his blocker.
The Hurricanes tied it little less than 3 minutes later when Skinner's shot from a sharp angle went under Neuvirth's blocker. "That first goal they scored was not a very good goal for my part," Neuvirth said.
Carolina dominated for long stretches of the second period, outshooting the Sabres 22-4. But the Hurricanes couldn't find a way to beat Neuvirth.
Carolina D Joni Pitkanen said his recovery from a broken heel was progressing and that he hoped to play next year. Pitkanen has missed the entire season.