After going 0-5-2 in their first seven games against the Blue Jackets, it took 26 saves from Rick DiPietro for a 4-0 Islanders victory on Saturday night.
"We're happy with the progress we've made, but we still have a lot of work to do," DiPietro said after getting his third shutout this season.
New York's penalty-killing was perfect, as it stopped all nine Blue Jackets power plays.
"Our penalty-killing was awesome," DiPietro said. "That was one of the biggest reasons we won tonight."
Fredrik Norrena made 18 saves for the Blue Jackets, who have alternated wins and losses in their past five contests.
Columbus controlled the puck early in the game. However, after Sergei Fedorov rang a shot off the post, the Islanders came back on a 3-on-1 break and took a 1-0 lead on Blake's 20-foot wrist shot 2:02 into the game.
"Those are the breaks, sometimes," Columbus coach Ken Hitchcock said. "We hit the post and it stays out. Then they come down and nick the post and it goes in."
Blake, playing in the final year of his contract, leads the team with 20 goals.
"I'm just working hard every shift," Blake said. "Playing with Alexei helps. He's an elite player. He's either going to set me up with a pass, or shoot the puck himself."
Islanders coach Ted Nolan was impressed with Blake from the start of training camp.
"One thing I noticed right away was that [Blake] competes hard," Nolan said. "He's got 20 at this point. Who knows how many he'll get?"
The Islanders are 12-0-1 when leading after one period, and the Blue Jackets dropped to 0-11-0 when trailing after 20 minutes.
Hunter made it 2-0 for New York at 6:33 of the second period. His one-timer from the lower left circle beat Norrena high to the short side.
DiPietro's toughest save came during a power play just after the goal, as he made a kick save on Duvie Westcott's screened slap shot.
Sillinger connected for a power-play goal at 2:41 of the final period, and Kozlov capped the scoring with his 13th goal at 9:01.
Nolan said he changed his strategy on the penalty-killing unit.
"We made some alterations on the penalty kill," Nolan said. "We were much more aggressive."
"The biggest problem was we had no entries," Hitchcock said. "We need to enter their zone to be able to do anything."
The Blue Jackets are frustrated with their past few road games. They were being shut out in back-to-back contests and have allowed 15 goals in their past three games away from home.
"Some games, we show up for, and then we don't show up for games like tonight, and that's something we definitely have to work on," Columbus forward Rick Nash said. "We have to have more consistency. They took it to us right from the start of the game."
"I think we have to grow up as a team," Norrena said. "To be a playoff team, we have to show up every night."
Columbus made only its third appearance at Nassau Coliseum, after winning the two previous trips. ... DiPietro has nine career shutouts. ... Islanders forward Arron Asham was back in the lineup after missing two games with a wrist injury. ... Blue Jackets defenseman Adam Foote needs five more games to reach 900 for his career. ... The Islanders are perfect this season on the second night of back-to-back games, raising their mark to 6-0-0.