Kaapo Kakko, Vincent Trocheck and Ryan Lindgren each had a goal and an assist, and Chris Kreider had two assists for the Rangers (30-14-8), who have won four straight and five of their past six games. Igor Shesterkin made 26 saves.
"Just from meeting [Tarasenko] this morning, he's really excited and it's a great way to start his career with the Rangers, scoring on his second shift," New York coach Gerard Gallant said. "He's a great hockey player. He's going to help our team."
Oliver Bjorkstrand had a goal and an assist, and Vince Dunn had two assists for the Kraken (29-18-5), who have lost the first three of a five-game road trip that continues Sunday at the Philadelphia Flyers. Martin Jones made 22 saves.
"We don't want to lose games like that in a row; it's something we've got to figure out," Bjorkstrand said. "We've got to go to Philly and make sure we get a win. It's a must-win, in my opinion. You just can't lose too many games in a row. It's not good for the team."
Kakko made it 2-0 with his first goal in 17 games at 5:01 of the first, and Trocheck extended the lead to 3-0 on a backhand at 14:29.
"It's tough to come back in a game when you're down 3-0 in the first," Seattle forward Jordan Eberle said. "We were slow. We gave them everything they had. They're fast and they're going to counter. I like the way we battled back, but ultimately it's tough to come back from three down."