Jiri Novotny, Jaroslav Spacek and Chris Drury also scored for the Sabres, who won for the sixth time in eight games. Martin St. Louis had two goals for Tampa Bay, which lost its second straight. It was the fifth time the Sabres scored six or more goals this season, and the second time in five games that they had seven.
Ryan Miller made 24 saves to win for the first time since Nov. 2. He missed four games in the middle of the month with an abdominal strain.
St. Louis gave the Lightning a 1-0 first-period lead before the Sabres broke loose for seven goals on 13 shots in the middle frame against Lightning goalie Marc Denis, making his second start in seven games.
Gaustad, who also added an assist, opened the scoring on the power play for Buffalo, fighting off Tampa Bay defenceman Luke Richardson along the side of the net to knock the puck in after it bounced off the end boards off a shot from Maxim Afinogenov.
Novotny scored 63 seconds later, and Vanek made it 3-1 at 7:48 of the second.
Gaustad and Spacek followed with goals 24 seconds apart in the middle of the period before Vanek pushed Buffalo's lead to 6-1 with his team-leading 15th goal at 14:02 while the Sabres enjoyed the man-advantage.
Drury capped the spurt with his 12th of the season with 2:41 to go.
The game was halted with 2:08 remaining in the second after the end boards partially split where the glass slides into it, and the teams were sent to their dressing rooms.
St. Louis made it 7-2 with his team-leading 13th of the season at 1:52 of the third.
Notes: The game marked the beginning of a three-game homestand for the Sabres. ... Buffalo D Toni Lydman missed his sixth straight game because of a neck strain. ... Tampa Bay captain Tim Taylor missed his third game with a neck injury. ... Coming in, the Lightning and Sabres were tied for the most goals scored in the second period (30). ... Sabres D Michael Funk got his first career NHL point with an assist on Spacek's goal.