Zibanejad and Vitali Kravtsov scored, and Shesterkin made 34 saves for New York (11-9-5), which fell to 4-5-4 at home this season, including 0-2-1 in its past three games. The Rangers are 1-4 in overtime games.
"It wasn't good enough," New York coach Gerard Gallant said. "I mean, we had two chances in overtime, but we didn't play well enough to even be there, to be honest with you. We were fortunate to be 2-2, but it wasn't good enough, especially the first two periods."
Tkachuk had an assist on
Tim Stutzle's
goal, a team-high eight shots on goal and a spirited fight with Rangers defenseman Jacob Trouba at 14:58 of the second period.
"Outstanding," Ottawa coach D.J. Smith said of Tkachuk. "Just stayed with it all night. Could have scored in the first. Stayed with it, physical. Obviously, the fight, that's a big boy fight. And then scored. There's not much more you can ask of a guy."
New York has lost four of its past five games (1-3-1), including two in which it held a lead in the third period.
The Rangers were up 3-0 against the Edmonton Oilers going into the third on Nov. 26 but lost 4-3 in regulation. They also led 2-0 in the first period against the New Jersey Devils on Monday but gave up the next four goals in a 5-3 loss.
"Just feels disappointing right now," Zibanejad said. "I didn't think we played good enough in the first or second. We did a little bit of a better job in the third. But it's definitely disappointing to do that, especially with [Shesterkin] giving us a chance to win."
Zibanejad gave the Rangers a 2-1 lead at 6:03 of the third, scoring from the front of the left post off a rebound after Braden Schneider's initial shot hit Artem Zub in the face, requiring stitches for the Senators defenseman.