Miller also stopped Martin St. Louis on a first-period breakaway. The Sabres, who had a seven-game winning streak halted in a 4-1 loss at Florida on Thursday, improved to 16-2-1 in their last 19 games.
Buffalo hasn't lost consecutive games since a four-game skid Nov. 2-9.
"That's something we take pride in," Miller said. "It's just implied now. I don't even think we have to mention it. We wanted to come back and just play hard."
Tallinder had his first multipoint game since Jan. 24, 2003, against Toronto.
"It's always hard to come here and play," Tallinder said. "Tampa has a great team, and to get two points here you've really got to be satisfied with that."
"Tonight, I thought we got our legs back underneath us," Sabres coach Lindy Ruff said. "We really skated. Our defense joined in on several of our goals and I thought that made the difference in the game."
Vaclav Prospal scored for the Lightning, who have lost four in a row and five of six. Tampa Bay trails the Southeast Division-leading Carolina Hurricanes by nine points.
"I think there were things within our team tonight amongst ourselves where it was boiling points and frustration with one another," Tampa Bay coach John Tortorella said. "I think it's very important right now, when it gets to that level, that we've just got to stay together. There are some tough times in a long season. This is the time we need to stay together and just get through it."
Tampa Bay played without center Vincent Lecavalier, who was given a one-game suspension for a high-sticking incident Thursday against the New York Rangers. Lecavalier has 14 goals and 35 points in 35 games this season.
The Lightning went 0-for-7 on the power play. Goalie John Grahame, who stopped 27 shots, has lost four in a row following a team record nine-game winning streak.
"We have to rally together," Tampa Bay captain Dave Andreychuk said. "We can't feel sorry for ourselves because of the situation we're in. There's no finger pointing. It's all of us together."
Prospal gave the Lightning the early lead, scoring 2:17 in. Buffalo tied it on Vanek's power-play goal at 6:15.
Drury scored from close range to put Buffalo ahead at 6:41 of the second. Jochen Hecht set up the goal, passing through the slot.
Tallinder made it 3-1 with 9:56 left in the second with his first goal in 15 games.
"We're just fragile right now," Tampa Bay center Brad Richards said. "Wish I could explain why."
Kotalik scored an empty-net goal with 2½ minutes left.
"This is a fun team," Miller said. "When we work hard and we play as a team, we get a lot of great opportunities to win. We've been cashing in on them."
- Lightning defenseman Pavel Kubina missed his second consecutive game with a groin strain.
- The Sabres were without C Daniel Briere (abdominal) and D Dmitri Kalinin (broken finger).
- Buffalo rookie LW Daniel Paille assisted on Tallinder's goal for his first NHL point, coming in his second game.
- Richards had his eight-game point streak end. He had two goals and 11 points during the stretch.