Adam Fox scored the go-ahead goal in the third period for New York (6-2-1), which has not lost on the road since being defeated 5-1 by the Washington Capitals on Oct. 13 in its season opener.
Shesterkin has allowed one or fewer goals in five of his past six starts, including a 31-save shutout in a 4-0 win against the Columbus Blue Jackets on Friday.
"We can sing his praises for four days," Fox said. "But I mean, he keeps us in every game and you never feel like you're out, ever. It's a tie game, and he keeps it that way. And obviously, we got one, so I can't say enough good things."
Jordan Eberle scored for Seattle (3-5-1), which went 2-2-0 on its first homestand. Philipp Grubauer made 15 saves in his fourth straight start.
"You're gonna win some games you should lose, you're gonna lose some games you should win, and I think this one was one of them," Grubauer said. "… We didn't capitalize on our chances, but overall I think we did a pretty good job."