Palat got his 200th NHL assist and 300th point, and Andrei Vasilevskiy made 28 saves for Tampa Bay (12-7-2).
"We played a pretty responsible game; we didn't make turnovers, we know they played Sunday]," Palat said. "We just wanted to put the puck deep and have a good forecheck. We did that in the second and third periods."
***[WATCH: [All Sabres vs. Lightning highlights]*
Rasmus Dahlin sustained a concussion in the second period for Buffalo (11-10-3), which won 5-2 at the Florida Panthers on Sunday but is 2-8-2 in its past 12 games. Sam Reinhart and Jimmy Vesey scored, and Carter Hutton made 23 saves.
Dahlin left the game after being hit in the left face-off circle by Erik Cernak with 2:34 remaining in the second.
"Very disappointing," Sabres coach Ralph Krueger said after announcing Dahlin's injury. "We'll let the League deal with the judgment of what happened. Right now, we're just worried about Rasmus and getting him healthy and hoping it's not a long-term injury."
Tampa Bay put the game out of reach in the third period with two shorthanded goals in 49 seconds. With Kevin Shattenkirk in the box for tripping, Cedric Paquette scored on a breakaway at 8:17 to give the Lightning a 4-2 lead.
"I just saw [Anthony Cirelli] after a face-off had time on the ice, their [defense] didn't pinch," Paquette said. "I had some open space in the middle, and [Cirelli] made a great pass. Lucky."
Palat one-timed Mathieu Joseph's pass on a 2-on-1 at 9:06 to make it 5-2.
"It's his skating," Lightning coach Jon Cooper said of Palat. "Whatever he did in the summer, he had to gain a step, and he has. He's got the grit, the physicality, all that, but he's picked up a step and it's worked well for him. This is as good as I've seen him play."
Nikita Kucherov gave Tampa Bay a 1-0 lead 13 seconds into the game with a wrist shot from the edge of the right circle for his eighth goal of the season.