The hosts controlled the first period, coming closest in the 24th minute when Fabian Espindola headed Thierry Henry's cross against the crossbar.
Henry was a constant threat for Red Bulls and could have had a first-half hat-trick were it not for the heroics of United goalkeeper Bill Hamid.
The home side continued to press in the second half, with Hamid denying Henry on several more occasions and doing well to keep out an effort from former Everton man Tim Cahill just after the hour mark.
United's best goalscoring opportunity fell to Chris Pontius in the 59th minute, but he was unable to find a way past Luis Robles between the home sticks.
Brandon Barklage and Cahill both had stoppage-time efforts saved as United withstood intense pressure to hold out for a draw.