Valteri Filpulla and Eric Brewer also scored for the Lightning, who have won five in a row. Ben Bishop stopped 27 shots.
Clarke MacArthur, Erik Karlsson, Ales Hemsky and Milan Michalek scored for the Senators. Robin Lehner had 23 saves while losing his sixth straight game. Ottawa is 0-3-2 in its last five.
After Callahan put the Lightning up 5-3, Michalek deflected Karlsson's shot from inside the blue line with 2:53 to go.
Ottawa then had a great chance to tie it as Callahan took a penalty with 55 seconds remaining. However, with Lehner on the bench for an extra skater—giving the Senators a two-man advantage—they were unable to beat Bishop.
Brewer scored his first goal in 95 games to give the Lightning a 4-3 lead at 3:48 of the third with a great shot off his backhand to sneak the puck in under the crossbar.
The Lightning took a 3-2 lead at the five-minute mark of the second period on a power-play goal by Filpulla, who knocked home a rebound just out of Lehner's reach.
Hemsky tied it after displaying some great puck handling before beating Bishop late in the second.
The Senators ended the first period tied 2-2 after getting off to a horrible start. Purcell, who hadn't scored in his last 19 games, ended his drought just 23 seconds into the game.
Ottawa managed to tie it on MacArthur's short-handed goal midway through the period. It was MacArthur's 21st of the season, tying a career high.
The Lightning regained the lead on Purcell's second of the night as he managed to dig the puck out through traffic in the crease and then shoot it in off the post.
Karlsson rounded out the scoring with a power-play goal, beating Bishop with a shot from just inside the blue-line.
NOTES: G Craig Anderson (upper body injury) and G Nathan Lawson (lower body injury) missed the game for Ottawa, and D Patrick Wiercioch and LW Matt Kassian were healthy scratches. ... The Lightning signed free-agent forward Cody Kunyk to a one year, two-way NHL contract. Kunyk, 23, skated in 37 games for University of Alaska-Fairbanks this season, posting 22 goals and 43 points. ... D Keith Aulie, D Mark Barberio, C Tom Pyatt and RW Nikita Kucherov were healthy scratches for Tampa Bay.