In the second half, Teal Bunbury almost doubled his side's lead when he latched onto a loose ball and fired a volley at goal, but his strike went inches wide of the post.
However, Bunbury did get on the scoresheet minutes later when he finished an tidy Revolution move in which Scott Caldwell and Lee Nguyen exchanged passes before the striker powered a shot past Robles.
Davies had the chance to put the game beyond the reach of New York as he burst through the middle, but he put he could only poke his effort past the post under pressure from Kemar Lawrence.
The Red Bulls got back into the match when Sacha Kljestan raced through the Revolution defence and round Shuttleworth to slot the ball into an empty net.
Bradley Wright-Phillips should have equalised for the visitors when he met a cross from the right with a powerful header, but his effort sailed over the bar as New England saw out the victory.