Ryan Shannon's short-handed goal 12:27 into the third spoiled Dwayne Roloson's shutout bid and drew Ottawa within 2-1. Nick Foligno tied it with his 12th goal at 16:05.
Spezza lost control of the puck on a breakaway but the puck trickled through Roloson for the winner.
Curtis McElhinney stopped 34 shots in his second straight start for Ottawa, his fourth in five games. McElhinney, who was claimed on waivers from the Lightning on Feb. 28, made 33 saves in a 3-1 win over New Jersey on Thursday night.
Vinny Lecavalier and Dana Tyrell scored for Tampa Bay, which is on a 2-4-4 skid.
Roloson made 31 saves, including a sprawling stop on Chris Neil late in the second.
The 41-year-old goalie appeared destined for his fifth shutout in 27 games with Tampa Bay since he was acquired in a Jan. 1 trade with the New York Islanders until Shannon beat him on a deke with 7:33 left in regulation.
Roloson denied Bobby Butler with another huge stop moments later but, Foligno tied it with 3:38 left.
Steven Stamkos, who leads the NHL with 43 goals, drove a slap shot off the post on a Lightning power play in the first.
Tampa Bay is fifth in the Eastern Conference with 89 points, one behind Pittsburgh.
Roloson faced just three shots in the first period, including a pair in the final 28 seconds of the opening period.
Nate Thompson set up Lecavalier's goal with a pass that found the Tampa Bay captain behind the Ottawa defense. Lecavalier drove in alone and fired a wrist shot past McElhinney 6:32 in.
Stamkos came close to making it a two-goal lead with a drive off a faceoff seconds after Senators right wing Butler was sent off for hooking at 8:59.
Tyrell made it 2-0 with his sixth goal 7:42 into the second.
NOTES: Roloson had no shutouts and a 6-13-1 mark in 20 games with the Islanders despite maintaining a 2.64 goals-against-average and a .916 save percentage. ... McElhinney dressed as Roloson's backup for two games from Feb. 25-27 but did not see any action with the Lightning after he was acquired from Anaheim for G Dan Ellis. ... Senators D Sergei Gonchar left in the second period with an upper body injury. He did not return.