After St. Louis pulled the Lightning even at 14:09, Palat made it 3-2 from the low slot 1:46 later.
Tyler Johnson had the other Tampa Bay goal as the Lightning (7-3) completed a seven-game homestand with a 5-2 record.
"It's awesome," Palat said.
Among the wins during Tampa Bay's season-long home stretch was a victory over the defending Stanley Cup champion Chicago Blackhawks.
"Are we a good team? I don't know," Lightning coach Jon Cooper said. "I'd say we're a resilient team. Right now, we're finding ways to win the close ones. And regardless of who they're playing, they're not getting rattled. It's just not happening to this team."
The Lightning overcame a strong performance by Buffalo goalie Ryan Miller, who stopped 36 shots.
"It's all about finding ways and we did that tonight," St. Louis said.
Brian Flynn and Thomas Vanek scored for Buffalo (2-10-1), which was coming off a 3-1 win at Florida on Friday night in which the Sabres were outshot 45-21.
"It's disappointing," Vanek said. "We battled hard and walked away with nothing. It doesn't take many mistakes with those guys to bury one."
Buffalo took a 2-1 lead when Vanek scored from the low left circle after taking a nifty pass from Marcus Foligno 12:46 into the third. Vanek has 24 goals and 39 points in 33 games against Tampa Bay.
Flynn tied it at 1-all with 7:02 left in the second when he scored off a pass through the crease by Cody McCormick.
Johnson put the Lightning up 1-0 on a rebound power-play goal at 12:48 of the first. The goal came after Sabres defenseman Tyler Myers failed to swat away the puck after Miller made a save on Valtteri Filppula's shot.
Buffalo has been outscored 16-1 during the first period this season.
Tampa Bay's Ben Bishop stopped a backhand by Steve Ott during a breakaway midway through the first. He finished with 20 saves.
Miller made several nice saves early in the second when Tampa Bay had two power plays. The goalie made a post-to-post save on St. Louis' right circle shot.
"You use your penalty killers, but then your offensive guys struggle to get into it and can't get in the rhythm, so it makes it really hard for them to produce some offense," Flynn said. "We need to be more disciplined to win hockey games like this."
Tampa Bay went 1 for 6 on the power play. The Sabres failed to score on three opportunities.
Lightning C Steven Stamkos had an assist, giving him six assists and 13 points over his past seven games. ... Buffalo LW Ville Leino, activated before the game from injured reserve, had an assist. He missed 11 games due to a rib injury. ... St. Louis has six goals and 13 points over his past nine games. ... The Sabres placed D Mike Weber on IR with an upper-body injury. ... Tampa Bay D Victor Hedman, who took a shot off his right foot in Thursday's win over Chicago, played.
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TAMPA, FL - OCTOBER 26: Teddy Purcell #16 of the Tampa Bay Lightning during warm ups prior to tonight's game...