Lloyd Sam ended New York’s 200-plus minute goalless drought but second half goals by Marco Pappa and Barrett secured a 2-1 Sounders victory.
In the only meeting between the two sides this season, referee Jose Carlos Rivero brandished four yellow cards in what was a physical contest.
The win keeps Seattle atop the Western Conference on 26 points with a game in hand over Vancouver Whitecaps (23 points) and ahead in the race for the Supporter’s Shield while Red Bulls stay third in the East with two games in hand.
Midfielder Cristian Roldan narrowly gave a patient Sounders team the lead after 12 mintues but his header was pawed away by Luis Robles.
The hosts who held 57 per cent of the first half possession were then undone against the run of play when Sam bundled home a header after the initial free kick deflected off the wall.
It was the first goal allowed by the league’s top defensive side since May 10 who have conceded just 9 times in 13 matches.
New York’s manager Jesse Marsch felt his side deserved a penalty when American international Brad Evans cleared out Bradley Wright-Phililps from behind in the box in the 54th minute.
A quarter-hour after the restart, Red Bulls had a chance to double their lead but Mike Grella’s low cross was just out of the reach of a stretching Sam.
Ecstatic to get the come from behind W and take 7 of 9 from this homestand and keep 1st. The Support was amazing all 3 games! #soundersfc
— Chad Barrett (@CBarrett19) June 1, 2015
Pappa then drew Seattle level on 69 minutes when he pushed his defender back before striking the ball into the net.
New York had done well to keep the potent attack of Obafembi Martins and Dempsey in check but the latter turned a cross goal bound that was poked over the line by second half substitute Barrett 30 seconds into stoppage time.
Seattle: Frei, Evans (Scott 75′), Mears, Marshall, Remick, Rose, Alonso, Roldan (Neagle 64′), Pappa (Barrett 89′), Dempsey, Martins.
Unused subs: Perkins, Gonzalez, Kovar, Azira.
New York: Robles, Ouimette, Duvall, Perrinelle, Lawrence, Martins, Sam (Abang 92′), McCarty, Grella (Richards 76′), Kljesten, Wright-Phillips.
Unused subs: Reynish, Miller, Lade, Obekop Davis.
Bookings: Perrinelle, Martins, Sam.
Referee: Jose Carlos Rivero