The visitors took the lead in the ninth minute when Federico Higuain was played behind the home side's defence and rounded Raul Fernandez. His initial effort was cleared off the line by George John but his clearance hit Higuain and rolled into the net.
The home side got back into the game in the 19th minute when Blas Perez coasted into the penalty area down the left and blasted a low shot past Matt Lampson.
Just before the break, Oduro was played one-on-one with Fernandez and made no mistake in slotting the ball into the bottom corner with his right foot.
After half time, Dallas got themselves back in the match again as they took advantage of confusion in the Crew's defense and dribbled through a crowd of players before tapping the ball home.
However, the visitors secured the win late in the game when Bernardo Arnor volleyed the ball into the roof of the net after it had fallen kindly into his path.