The defending MLS champion Sounders (1-1-1) received goals from Clint Dempsey, Jordan Morris and Harry Shipp.
The Red Bulls slipped to 2-1, and the loss ended their 18-game unbeaten streak in the MLS regular season, dating back to July of 2016. The all-time MLS record is 19 games, shared by Columbus and FC Dallas.
Morris gave the Sounders the 2-1 lead in the 66th minute from Nicolas Lodeiro and Joevin Jones. Lodeiro received a pass from Jones from the end line and sent a cross in the air that Morris was able to head in to the left of the Red Bulls net.
Shipp scored in the 79th minute for Seattle.
Bradley Wright-Phillips struck in the 57th minute on a header from the middle of the box on a cross from Sal Zizzo. It was the first marker of the season for Wright-Phillips, the Golden Boot winner each of the last two seasons.
Dempsey scored on a penalty kick in the 28th minute after Red Bulls keeper Luis Robles cut down Morris in the box. Morris had intercepted Red Bulls' defender Damien Perrinelle's ill-advised back pass to Robles.
The goal was the 34-year-old Dempsey's 61st in MLS. He missed the last quarter of 2016 with an irregular heartbeat.
Seattle keeper Stefan Frei had to make just one save to earn the victory. Felipe had the only shot on target for the Red Bulls in the first half.
Robles appeared in his 143rd consecutive league game.