Josh Norris also scored for the Sens (8-16-1) while Filip Forsberg made 25 saves for his first NHL shutout. Brian Elliott stopped 19 shots in the Tampa Bay (17-6-4) net.
"We just played the right way," Tkachuk said. "We were getting on the forecheck, doing the details really well tonight. We're going to build off of that."
Tkachuk's hat-trick was the third by a Senator this season after Drake Batherson and Zach Sanford. He was the first Sens captain to score a hat-trick since Jason Spezza on Oct. 15, 2013.
"That top line [and] some of the things Brady did in the third period, putting it in, changing, you can just see how he's learning to win in the NHL," Sens head coach D.J. Smith said.
In his 66th career NHL game, the 29-year-old Forsberg recorded his first NHL shutout.
"It's really nice to get a shutout for the confidence," Forsberg said. "Obviously, the most important part is that we played a really solid game for 60 minutes and we have to keep building from there."
The Sens had the only goal of the first and it came through Norris as Drake Batherson found him with a cross-ice saucer pass for the one-timer to bury his team leading 12th goal of the year.
The Lightning thought they had the tying goal with 52.1 seconds left in the period but it was waved off as Alex Killorn batted the puck past Forsberg with a high stick.
A pair of Tkachuk power play goals gave the Sens a commanding 3-0 lead in the second: His first came 3:32 into the frame as he received a nifty backhand pass from Tim Stützle out front and outwaited Elliott before sliding the puck five-hole.
His second came 1:59 later as he got a slight tip on Thomas Chabot's blast from the point shot to record his second goal of the game.
"As good as the power play was that was as good as our penalty kill has been," Smith said after his team killed off five Tampa Bay power plays. "
Tkachuk completed the hat-trick with 3:43 to play in regulation as he scored into the empty-net. It was the first time a Senator had scored a hat-trick against the Bolts and the 10th natural hat-trick in franchise history.
Chabot extended his point streak to four games with a pair of helpers while Batherson recorded three assists.
The Sens open a three-game road trip Tuesday when they visit the Florida Panthers. The team is back home Dec. 19 to host Boston.
