Tampa Bay was down 2-1 with 5:41 left in the third period when St. Louis netted his sixth of the season on a one-timer off a pass from Steven Stamkos for the equalizer.
Less than 2 minutes later, Palat dug out the rebound of a Tyler Johnson shot and put it past Ryan Miller for the go-ahead tally.
Johnson also scored for the Bolts, who improved to 7-3-0 on the season.
Ben Bishop stopped 20-of-22 shot in the victory.
Thomas Vanek and Brian Flynn each had a goal and Miller ended up with 36 saves for the Sabres, who fell for the fourth time in five games.