Shantz had two goals and Phil Housley added a goal and two assists for the Flames (3-7-2-0), who are 2-3-0 on a six-game road trip.
Bure set up power-play goals just 36 seconds apart in the second period by Housley and Shantz to tie the game 2-2.
Then, with the Flames trailing 3-2 midway through the third, Bure broke in on goaltender Ron Tugnutt and poked in his own rebound while falling. The goal was Bure's ninth, just one behind league-leading Brian Savage of the Montreal Canadiens.
"We knew when we traded for him last year he had big-time character," said Flames coach Brian Sutter. "He always plays the same."
In overtime, Bure carried the puck into deep into the Senators' end and threw it in front, where Shantz poked the puck under Tugnutt's pads 3:07 into the extra period.
Calgary outshot the Senators 37-28.
"They were finishing their checks, shooting pucks, driving to th net," said Roy, who scored Ottawa's first goal after picking up a loose puck in front of Grant Fuhr. "In overtime, they had wings."
After Arvedson beat Fuhr on a screened shot to give Ottawa a two-goal lead in the first period, Calgary took advantage of Senators penalties with power-play goals by Housley and Shantz just 36 seconds apart to tie the game.
Ottawa's penalty-killers entered the game ranked second in the league at 90.6 per cent, while Calgary's power play ranked 13th at 18 per cent. Calgary finished 2-for-2 on the power play and Ottawa was 0-for-3.
"We got a 2-0 lead and then we stopped playing," said Bonk, who gave the Senators a 3-2 lead with a breakaway goal at 15:33 of the second. "We didn't get going at all, we should have won that game.
"Now, we just have to forget it."
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