Brown was hauled down inside the area by Chivas midfielder Eric Avila in the 7th minute, triggering the penalty shot.
He converted with a volley inside the left post for the Rapids (2-3-2), who recorded their second straight win.
Carlos Alvarez had a good late chance for Chivas (3-2-1) late in the game but sent it over the net after coming in alone on Rapids goalkeeper Clint Irwin.
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