Anders Lee added two goals for the Islanders, who snapped an eight-game losing skid (0-4-4).
Gibson, starting just his third game this season and fifth of his career, was under siege the final two periods but held strong enough at the Scotiabank Saddledome.
With a pair of goals 28 seconds apart in the early going, the Islanders made it a horrible return for Flames goalie Mike Smith, who missed 13 games due to injury.
New York's Nick Leddy opened the scoring just 2:14 into the game, wiring a long point shot off the crossbar and in after Casey Cizikis cleanly won a faceoff in the Calgary zone.
Boychuk doubled the lead at the 2:32 mark, giving the visitors two goals on three shots. Boychuk's point shot went off the stick of Flames forward Troy Brouwer and beat Smith's glove hand.
Johnny Gaudreau gave the Flames a jolt of life with his first goal in eight games at 7:23 of the opening period. He buried a short-side, top-corner shot from bottom of the left circle, but the Islanders bounced right back.
Jordan Eberle made it a 3-1 game at 10:25 of the first period by showing a goal-scorer's touch. Smith stopped Eberle's redirection of Boychuk's slap-pass, but Eberle grabbed the rebound and hit the mark.
Lee then made it a 4-1 tilt 50 seconds into the second period. Smith kicked out a juicy rebound after stopping Boychuk's point shot, and Lee immediately blasted his chance into the net.
Calgary captain Mark Giordano scored at 7:24 of third period to make it a 4-2 game. Giordano touched the puck last during a wild scramble that finally ended with the puck in the net, giving him three goals in the past four games.
Lee rounded out the scoring with an empty-netter, his team-high 33rd goal of the season, with 10.1 seconds remaining.
Gaudreau, who has six points in a four-game point streak, has 79 points on the season, a career high.
Smith stopped 22 shots for the Flames.
--Field Level Media
NY Islanders | Calgary | |
Johnny Boychuk 3 | Points | Johnny Gaudreau 2 |
Anders Lee 2 | Goals | Johnny Gaudreau 1 |
Johnny Boychuk 2 | Assists | Johnny Gaudreau 1 |
N/A | Power Play Goals | N/A |
N/A | Short Handed Goals | N/A |
Christopher Gibson .962 | Save Percentage | Mike Smith .846 |
Christopher Gibson 50 | Saves | Mike Smith 22 |
Team | Shots | Goals | Power Play | Penalty Kill | Penalty Mins | Face Offs Won |
NY Islanders | 27 | 5 | 0-1 | 4-4 | 13 | 30 |
Calgary | 52 | 2 | 0-4 | 1-1 | 7 | 42 |