Josh Norris scored for the Senators (22-28-5), who had won three in a row, including 4-2 at the Winnipeg Jets on Saturday. Anton Forsberg allowed four goals on 23 shots before being replaced by Filip Gustavsson, who made 11 saves.
"They're a team that needs to win out," Ottawa forward Brady Tkachuk said. "They came out ready to play. We didn't. It's unfortunate. It's a little frustrating. We know we're going to be ready for our last game of the year (against the Toronto Maple Leafs on Wednesday). It's not only about ourselves, but making our community proud. They've been sticking with us through thick and thin. We just can't give them that effort we gave them tonight."
Gaudreau scored on a backhand deke on a breakaway to put Calgary up 1-0 at 10:39 of the first period after a stretch pass from Mark Giordano.
"I saw him have a little bit of some time and space down low," Gaudreau said. "The D were pretty separated. I kind of just split up the middle and he fired it right up to me and we found a way to get the lead."
Gaudreau has scored 15 points (six goals, 10 assists) in the past 12 games.
Mikael Backlund scored short side over Forsberg's shoulder 17 seconds into the second period to make it 2-0, and Tkachuk gave the Flames a 3-0 lead when he scored off a backdoor pass from Gaudreau at 4:38.
"He put it right in a perfect spot for me right where the D couldn't get it," Tkachuk said. "Just tried to push the D off to try to give him a little bit more space, but the pass was right on the tape. I don't even know if I needed to do that. It was a great play, and that put us up by three, so it was a good lead that we needed to have at that point."
Michael Stone scored on a slap shot off the rush to make it 4-0 at 10:05. Forsberg was pulled after the goal.
Norris cut it to 4-1 at 11:06 with a chip shot.
"They played a playoff game," Senators coach D.J. Smith said. "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 and they played a great game and they gave us nothing wire to wire."