"Reto was simply outstanding. He was the guy who kept us in the game and gave us a chance," Flames coach Bob Hartley said. "He made big saves early and gave us a chance. Then we started skating better and executing better."
Jones banked the puck off the Islanders goalie Evgeni Nabokov from a sharp angle at 14:28 to lift the Flames to their first win at Nassau Coliseum since 2004.
Mikael Backlund added an empty-net goal for the Flames, his 14th of the season.
The 27-year-old Berra, who will join Anaheim's Jonas Hiller with Switzerland's team at the upcoming Winter Games in Sochi, improved to 8-15-2 and won for the first time in regulation.
The Islanders went on the power play with three minutes left when Jiri Hudler was called for hooking but couldn't muster much of an attack.
Monahan's team-leading 16th goal opened the scoring at 14:15 of the second when he wristed a loose puck from outside the crease over Nabokov's left shoulder. The 19-year-old rookie center was the sixth overall pick by the Flames in last June's draft.
Wideman made it 2-0 at 18:08 with a snapshot from the slot, his fourth of the season.
Hickey cut the deficit in half at 19:53 of the second, snapping the puck past Berra for his fourth of the season.
It was the first point in five games for Tavares, who is second in league scoring with 23 goals and 42 assists, 12 points behind Pittsburgh's Sidney Crosby. The loss left Tavares exasperated once again.
"We seem to find more way to lose than win," he said.
The Flames had only 22 shots but made them count. Calgary (22-28-7) finishes its road trip with a visit to Philadelphia on Saturday before the Olympic break. Calgary swept a five-game homestand before losing at Montreal 2-0 on Tuesday.
"We worked hard and earned out chances," said center Matt Stajan, who was originally credited with the go-ahead goal. "We didn't give up, kept the pressure on them and it paid off for us."
The Islanders (22-29-8) were within five points of a playoff spot when they beat the Rangers 5-3 on Jan. 21 but six losses in seven games have essentially ruined their postseason hopes.
After the game, the Islanders traded forwards Peter Regin and Pierre-Marc Bouchard to the Chicago Blackhawks for a fourth-round pick in this June's draft. Regin, who was scratched against Calgary, had two goals and five assists in 44 games. Bouchard was playing for AHL Bridgeport after four goals and five assists in 28 games.
The Islanders were starting a five-game homestand before and after the Olympic break. They'll host Colorado on Saturday before the NHL season halts for the Winter Games at Sochi. They'll resume against Toronto on Feb. 27.
"We have to simplify our game and just get better with execution," Islanders coach Jack Capuano said.
The Islanders had won three straight and five of six against Flames, including the previous three on Long Island. The teams last met Dec. 29, 2011, when the Islanders won 3-1. The last time the Flames won at Nassau was Jan. 6, 2004.
The Islanders are 19-7-3 when they score at least three goals and are 3-22-5 when they do not.
The teams will play again March 7 in Calgary. ... The Islanders remained without defenseman Travis Hamonic (upper body). ... The Flames were missing goaltender Karri Ramo (knee) and forward Curtis Glencross (ankle).