Dempsey, who will sit out U.S. national team friendlies against Germany and the Netherlands this month as he awaits the birth of his fourth child, ghosted in at the far post behind the New York defence to meet a cross from the right.
His deft left-footed volley was headed for the net before Barrett got a touch on it and was credited with the goal.
The win lifted the Sounders three points clear of Vancouver at the top of the Major Football League Western standings.
Sunday's other game was a rematch of last year’s MLS Cup final with Los Angeles Galaxy drawing 2-2 with the New England Revolution in a match played in driving rain with all four goals coming in a wild first half.
Galaxy striker Robbie Keane missed the game as he recovers from a groin strain.
LA remain fifth in the West while Revolution stay second in the East.
(Reporting by Andrew Both in Cary, North Carolina; Editing by Peter Rutherford)
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