Vermette's second goal of the game with seven minutes to play in the third period stemmed the tide as the Senators watched a 4-0 first-period lead shrink to 4-2.
Patrick Eaves and Joe Corvo also scored for the Senators (14-12-1), who have now won seven of their past eight games and are now eighth in the Eastern Conference.
Eric Perrin and Vincent Lecavalier scored in the second period for the Lightning (13-12-2), who have lost three straight following a three-game winning streak.
After dominating the Lightning through the opening 20 minutes, the Senators were on their heels most of the second period, being outscored 2-0 and outshot 17-9.
Perrin and Lecavalier scored less than two minutes apart in the second to trim the Sens' lead in half. The Lightning had several chances to narrow the gap even further, but they squandered back-to-back power plays shortly after Lecavalier's goal.
Things started poorly for the Senators as Peter Schaefer was penalized for interference 1:33 into the game. Their fortunes quickly turned though, as Eaves stole the puck at the Lightning blue-line and beat Johan Holmqvist on the backhand for a shorthanded goal.
Corvo doubled the lead when his point shot beat Holmqvist on the power play at 6:49.
Vermette made it 3-0 fewer than two minutes later with a highlight-reel goal. Standing on the goal line facing the end boards, Vermette shot the puck from behind his back and between his legs into the short side past Holmqvist.
That goal spelled the end of the night for the Tampa netminder. He was replaced by Karri Ramo, who made his NHL debut.
Spezza gave the Senators their biggest lead at 16:21 after taking a feed from Chris Kelly, who stole the behind the Tampa net and found Spezza all alone in front.
Notes: The Senators set a club record for most consecutive goals without being scored upon. After falling behind 1-0 in Carolina Nov. 28, the Senators scored the next four goals in a 4-1 win then Shutout the Florida Panthers 6-0 Nov. 30. Their 4-0 lead Saturday gave them 14 straight goals.