Gavrikov scored on a rebound of Alexandre Texier's shot from the right circle for his second goal in as many games.
"There were a couple of lucky bounces for us, but it doesn't matter, we won," Gavrikov said.
WATCH: [All Blue Jackets vs. Islanders highlights]
Joonas Korpisalo made 35 saves for Columbus (17-14-6), which extended its point streak to eight games (6-0-2) heading into the Christmas break. Korpisalo has been the goalie of record in all eight (1.68 goals-against average, .942 save percentage) to set his NHL-best point streak.
"He gives the team confidence the way he is playing, and the team responds by playing really well in front of him," Blue Jackets coach John Tortorella said.
Mathew Barzal had a goal and an assist, and Thomas Greiss made 32 saves for New York (23-9-3), which has lost three straight at home (0-2-1) and is 1-2-1 in its past four heading into the Christmas break.
"I'm really disappointed. I thought we should have gotten a point," Islanders coach Barry Trotz said. "I take nothing away from Columbus; they're playing really well. They're missing some pieces from their team up front, but their young guys systematically are working very hard."
Anders Lee gave the Islanders a 1-0 lead at 6:17 of the first period after a pass from behind the net by Barzal. It was Lee's second goal in as many games.