Palmieri gave New York a 1-0 lead at 15:45 of the first period. He scored from the bottom of the left circle on the rebound of Pierre Engvall's shot.
Brock Nelson almost extended the lead at 8:31 of the second period, but his shot hit the right post.
“Tonight, we did the best we could to keep them to the outside and limit their high-danger chances,” Islanders captain Anders Lee said. “They’re a skilled team, so they’re still going to get some, but [Varlamov] had a phenomenal game.
“That was a gritty road game from us tonight.”
Blue Jackets forward Jack Roslovic had three chances to tie it during the third period.
His first attempt after he got behind the defense on a rush was gloved down by Varlamov at 6:43. He then hit the left post on a partial breakaway during a power play at 8:26, and during the same power play, he got another partial breakaway at 9:05, but his backhand was stopped by Varlamov before Adam Pelech cleared the rebound off the goal line.
“They had two great chances, but [Pelech] helped me out when the puck was on the goal line,” Varlamov said. “He made the save. Another huge save by 'Pel.'”