Ottawa got a goal from Patrick Wiercioch.
St. Louis beat goalie Robin Lehner from just outside the crease to give Tampa Bay a 2-1 advantage 13:19 into the second. The Lightning captain tied Jason Arnott for 91st place on the NHL points list with 938.
Filppula's goal from the left circle made it 3-1 at 10:56 of the third.
St. Louis put the Lightning up 1-0 when he scored from in-close while falling forward with 3:15 left in the first.
Wiercioch tied it at 1-all with a power-play goal 6:11 into the second. The defenceman had not scored a goal in his previous 20 games dating back to last season.
Bishop made a nice save early in the first on Chris Neil's slot shot. The Lightning acquired Bishop from Ottawa last April.
Lehner kept the Senators within one early in the third when he stopped a shot by Nate Thompson during a breakaway.
The Lightning were already without several key players, including centre Steven Stamkos (broken leg), left wing Ryan Malone (broken ankle), and defencemen Eric Brewer (lower body) and Radko Gudas (upper body).
Stamkos has a goal of returning in February, while Malone will be out three to four weeks. Brewer and Gudas could soon return.
NOTES: St. Louis was heldless pointless in his previous three games. ... Ottawa C Kyle Turris had an assist and has seven assists and 12 points in 10 games against Tampa Bay.