Ilya Sorokin made 31 saves in his fifth shutout this season and No. 15 for his career. Samuel Bolduc, Simon Holmstrom and Zach Parise also scored in New York's fourth straight win, and Jean-Gabriel Pageau added two assists.
After Horvat's goal, the crowd gave him a standing ovation, greeting him with a "HOR-VAT! HOR-VAT!" cheer. He could savor the moment as he sat on the bench while his name was called by the public address announcer as the goal scorer.
"It's unbelievable," Horvat said in his postgame on-ice interview, saluting the fans. "Thank you, guys. It's pretty special, and I'm glad we got the win here for you guys tonight."
The Islanders acquired Horvat in a blockbuster trade with the Vancouver Canucks on Jan. 30. The All-Star center then agreed to an eight-year, $68 million contract on Sunday.
The 27-year-old Horvat skated in his first game with New York on Monday night, helping the Islanders to a 2-1 victory at the Philadelphia Flyers.
"It was pretty amazing," Horvat said of the chants that followed the goal. "I haven't had a chant like that in a building, that loud, I don't think ever, so that was pretty amazing."
Barzal concurred, and he appears to like playing on the same line with his new teammate.
"[He] just seems to be there in the right goal-scoring spot, so it's exciting," Barzal said. "I'm excited. I feel like I even got a little more juice out there just playing with him."
Horvat's career-best 32nd goal of the season gave New York a 4-0 lead.
Martin Jones had 23 saves for the Pacific Division-leading Kraken, but Seattle dropped the opener of a five-game trip to begin the second half of the season.
Kraken forward Andre Burakovsky left after his first shift with a lower-body injury.
Holmstrom made it 2-0 with a nifty forehand-backhand combination at 18:29. Pageau found the Swedish winger all alone in the slot from behind the net.
Parise masterfully redirected Pageau's shot past Jones to give New York a three-goal advantage 3:38 into the second.
New York improved to 22-5-2 when scoring at least three times, including shootout winners.
"There is a lot of belief in this group, a lot of energy, and you are seeing now a little bit of momentum," Islanders captain Anders Lee said. "It's a tough game [for Seattle] coming out of the break, so we were able to take advantage of that."
On Thursday, Horvat and the Islanders will host the Canucks, the team that just dealt the veteran forward to New York.
The Associated Press contributed to this report.