Tim Stützle took a stretch pass off the boards before moving across the blue line and sliding the disc to Tkachuk, who made no mistake, beating Hellebuyck with a high wrister to give the Senators the win.
Nick Cousins and Jake Sanderson also scored for the Senators.
Linus Ullmark turned aside 23 shots for Ottawa, who moved to 15-14-4.
Neal Pionk and Logan Stanley scored for the Jets, who picked up a single point with the loss.
Connor Hellebuyck made 28 saves for Winnipeg, dropped to 15-16-1.
Kurtis MacDermid hit the post with a wrister just over the five minute mark into the contest.
There was no scoring in the first period.
Nick Cousins opened the scoring 10:47 into the second, when he broke in and fired a wrister that Hellebuyck only managed to get a piece of with his blocker.
The goal was his 5th of the season.
Neal Pionk tied it up with a slap shot from just inside the blue line when he one timed a Kyle Connor saucer pass past Ullmark to make it 1-1.
The goal was his 2nd of the season.
Logan Stanley gave the Jets the lead at 18:35 when Gabriel Vilardi gained the blue line before sliding the disc to Morgan Barron who relayed it to Stanley as the towering defenceman fired a wrister past Ullmark from the top of the face off circle, beating him short side to make it 2-1.
Jake Sanderson tied it up late with a shot through traffic from the blue line when he converted a Tim Stützle feed.
The game went to overtime.
SHOTS-OTTAWA 31 WINNIPEG 25
ATTENDANCE-13,566
