Celebrating their first championship in their home opener at CenturyLink Field, the Sounders needed second-half goals from Jordan Morris and Harrison Shipp to get past the Red Bulls.
Clint Dempsey had opened the scoring from the penalty spot, but his effort was cancelled out by Bradley Wright-Phillips in the 57th minute.
However, Morris and Shipp saw Seattle to their first win of the campaign, lifting them into sixth in the Western Conference, while the Red Bulls' perfect start to the season came to an end.
Dempsey put the hosts ahead from the spot after Morris was brought down by goalkeeper Luis Robles, who was done no favours by a poor Damien Perrinelle back pass.
The equaliser came via Wright-Phillips, who headed in a cross from Sal Zizzo from the right.
Morris restored Seattle's advantage in the 66th minute, heading in a clipped Nicolas Lodeiro cross to make it 2-1.
Shipp scored the sealer, redirecting a Joevin Jones effort from range to complete the win.