The Islanders felt the effort had been there in a pair of hard-luck losses in California, so winning - and winning a low-scoring, grinding affair - was a confidence booster and felt like a just reward.
"I'm just glad with the effort that the guys put in that they were rewarded," head coach Jack Capuano said.
John Tavares and Sean Monahan scored in regulation, while Thomas Greiss made 25 saves for the win.
The Islanders came out with the start they wanted on Monday, skating fast, generating offensive zone time and taking an early lead. Tavares was set up by a nice, no-look backhand feed from Josh Bailey, which the captain accepted and roofed past Elliott at 7:28.
"It was about time, I've had some chances," Tavares said after busting a nine-game goalless drought. "Great play by Bails. I almost had too much time. I was able to read the goalie and pick the corner, so I'll try to build on it."
With Tavares' goal being the only marker for 40 minutes, Greiss was the Islanders' stopper, keeping the Flames at bay with some timely saves through two periods. He denied Deryk Engelland on a first period breakaway, staying with the Flames defenseman kicking out the right pad.
He was busier in the second period, making 11 saves, including another breakaway save - this time on Matt Stajan. Greiss was calm, cool and steady, gloving a Michael Ferland shot, controlling his rebounds and slowing things down when necessary.