The home side took the lead after 30 seconds when a slick attacking move which culminated in Ambroise Oyongo's shot being turned into the path of Wright-Phillips, who dispatched the rebound.
Lloyd Sam should have doubled New York's lead when he was teed up by Wright-Phillips, but he blasted his effort straight into the legs of Stefan Frei.
The Sounders continued to carve out opportunities of their own and Neagle drove towards the box before firing an effort at goal with his left foot, but Luis Robles made the stop.
In the second half, the Red Bulls were handed the chance to extend their lead when Sam was brought down in the area by Dylan Remick and a penalty was awarded. Wright-Phillips took the spot kick and drilled the ball past the outstretched hand of Frei.
Wright-Phillips completed his hat-trick in the 56th minute when he latched onto Oyongo's cross into the box and fired a first-time effort past Frei, leaving the keeper with no chance.
Seattle got a goal back almost immediately as Neagle got in behind the New York defence, and the forward slid a pass to Clint Dempsey, who was left with a simple finish to guide the ball past Robles.
However, the home side sealed their win when Tim Cahill scored their fourth goal of the evening, as he collected Oyongo's deflected cross and beat a defender before slotting the ball into the net via the post.