Willis made a tough save on rookie Deshorn Brown in the 12th minute and also stopped him in the 52nd, making a fingertip stop on a shot from just outside the six-yard box.
In the game's late stages Willis turned aside a pair of shots from Nick LaBrocca, enabling cellar-dwelling United (2-13-4) came away with a point.
The Rapids (7-7-6) used the draw remain one point out of fifth place in the Western Conference.
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