Jacob Markstrom made 29 saves, including 11 in the first period for the Flames (25-23-5), who had a four-game winning streak end.
"Jacob kept us in the game in the first period for sure," Calgary coach Ryan Huska said. "I thought we got a little bit better as the game went on, but their goaltender played well too tonight. He made some good saves at key times for them."
Calgary ended a four-game road trip coming out of the All-Star break with a 3-1-0 record, losing in New York after picking up wins against the Boston Bruins (4-1), New Jersey Devils (5-3) and New York Islanders (5-2).
"Obviously, we had a chance to sweep the road trip and put ourselves in a really good spot, and for whatever reason, it wasn't our night tonight," forward Blake Coleman said. " 'Marky' gave us a chance in a game we didn't deserve, but we'll take three wins on the trip. We won't sit on this one. We'll go home and take care of business."
Shesterkin had a rough January, going 4-5-1 with a 3.25 goals-against average and .863 save percentage in 10 starts.
He represented the Rangers at NHL All-Star Weekend in Toronto from Feb. 1-3, but coming out of the break, they gave him extra time to work with goalie coach Benoit Allaire by starting Quick in the first two games.
New York won each game, 2-1 in overtime against the Colorado Avalanche on Feb. 5 and 3-1 against the Tampa Bay Lightning two nights later.
Shesterkin got back in the net against the Chicago Blackhawks on Friday, making 28 saves in a 4-3 overtime win at United Center. He followed it with arguably his best performance of the season Monday.
"Everybody knew he was going to get back to how he plays," Rangers defenseman Jacob Trouba said. "I think every player goes through slumps, it's just more noticeable when it's the goalie and everybody's eyes see it more. But everybody goes through a stretch of games that they don't like. I would just say it's easier to hide at times when you're a [skater].
"He works hard in practice, took that time to reset and he was great tonight."
Cuylle gave the Rangers a 1-0 lead at 12:31 of the second period off a rebound of Kaapo Kakko's shot.