Mikael Backlund, Michael Stone, Dillon Dube and Mark Giordano also scored for Calgary, while Jacob Markstrom made 19 saves.
It was the ninth and final regular-season meeting between the North Division opponents.
It was a game the Flames had to win - and they did so in emphatic fashion, controlling almost every facet of play right from the opening faceoff.
They built up a 7-1 edge in shots (with some additional chances that missed the net) before Gaudreau opened the scoring at 10:39 with his 19th of the campaign.
Leading up to the tally, No. 13 feathered a pass to Tkachuk at the top of the crease but he couldn't get his stick on the bouncing puck in front of Ottawa goalie Anton Forsberg.
Then it was Gaudreau getting the puck in the high slot as he stepped into a shot that rang off the bar.
The Gaudreau-to-Tkachuk combo hooked up again, the latter's offering gloved by Forsberg.
Tkachuk then got a helper, passing the puck to Giordano deep in the Flames zone, who sent a long stretch pass to Gaudreau, the speedy forward stumbling slightly to slow him down a bit for a moment, but still able to turn it up and beat a chasing Victor Mete in a footrace, sliding the puck five-hole with the Ottawa defenceman in his hip pocket.
Johnny Hockey came in to the game with points in nine of his last 11 outings (5g, 7a).