Brian Elliott made 19 saves for the Lightning (17-6-4), who were shut out for the first time this season and had won five in a row. Elliott lost to the Senators in regulation for the first time (7-1-2).
The Lightning were 4-1-0 on a five-game road trip.
"When you're getting these road trips, whether it's three, four or five games, you've got to take more points than games played, and we did that," Tampa Bay coach Jon Cooper said. "If we lose the first game and win the next four, it's probably a way better feeling than it is right now. It [stinks] to do interviews like this right after the game. But for the big picture, those eight out of 10 [points], that's a good trip."
Josh Norris one-timed a saucer pass from Batherson to give the Senators a 1-0 lead at 12:40 of the first period.
Tkachuk scored through the five-hole on the power play to make it 2-0 at 3:32 of the second off a no-look pass from Tim Stutzle. He scored his second power-play goal of the game when he tipped a Thomas Chabot point shot to make it 3-0 at 5:31.
Tkachuk missed the first three games of the season after signing a seven-year contract as a restricted free agent Oct. 14. He was named Ottawa captain on Nov. 5 and has scored 17 points (nine goals, eight assists) in 15 games since.
"I would say probably two weeks ago, you could see his legs were under him, and then because his legs are under him his hands are now under him," Smith said. "And on top of that, he's mean, he plays with emotion and now he's making the right plays, making heavy plays, making smart plays, and he's driving us right now up front."