Mikael Backlund, Michael Stone, Dillon Dube and Mark Giordano also scored for Calgary. Jacob Markstrom stopped 19 shots for his 19th win.
The Flames moved eight points behind Montreal for the fourth and final playoff spot in the North Division. To make the postseason, Calgary needs to win its remaining four games against the division-worst Canucks and the Canadiens needs to lose its remaining two games against Edmonton in regulation.
“We put ourselves in a terrible position, to be honest,” Tkachuk said. “We just have to win and pray. That’s all we can do tomorrow (Monday). … Never thought we’d root for Edmonton, but we need them to win these.”
Gaudreau echoed the sentiment.
“I’ll check out the scores,” he said. “Probably won’t watch it. Just sit there and hope Connor (McDavid) can do his thing.”
Josh Norris scored the lone goal for Ottawa, which lost in regulation time for just the second time in its last 11 games. Anton Forsberg gave up four goals on 23 shots. He was replaced by Filip Gustavsson, who made 11 saves in relief.
Knowing they were facing elimination, the Flames played desperate hockey from the opening faceoff.
After Gaudreau hit the post in the opening minutes, the Flames forward was not denied on his next opportunity. On a fast break, Gaudreau took a stretch pass from Giordano right on the tape and slid a delicate backhanded shot between Forsberg’s pads while falling to the ice at 10:39 for his fourth goal in his last six games.
Meanwhile, the Senators, who were coming off a 4-2 win over the Winnipeg Jets on Saturday, could not get anything going in the first. They went more than 15 minutes without firing a puck on Markstrom and were outshot 10-3.
“First period was better,” Flames coach Darryl Sutter said. “We didn’t give up many chances in the first period. That’s the way you’ve got to play. Our team has to play like that to win.”
Calgary opened the floodgates in the second period, scoring three times by the halfway mark.
Backlund put the home team up 2-0 just 17 seconds in to cap off a good cycle play in the offensive zone.
The Flames added to their lead at 4:38 when a patient Gaudreau found Tkachuk in the slot for his first goal since April 2.
At 10:05, Calgary made it 4-0 on its 23rd shot when Stone’s slap shot from the point blasted past Forsberg, chasing the Senators’ goalie from the game.
“They played a playoff game,” Senators coach D.J. Smith said of the Flames. “They gave us no room. They know that they can’t afford to give up a single point or they’re out. They jumped all over a team that played last night. … Tonight we just had too many guys not going.”
Gustavsson started the contest on the bench following his 27-save performance on Saturday.
That goaltender change may have briefly sparked the visitors as Norris scored a minute later on a beautiful backhand pass from Drake Batherson.
But the Flames added two more in the third period to put the game out of reach. Dube sealed it on the rush with a one-timer at 6:45 before Giordano made it 6-1 at 13:53.
Calgary finished 3-5-1 against the Senators this season.
Senators: Host Toronto on Wednesday night in season finale.
Flames: Host Vancouver on Thursday night.