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.
Ray Liotta sues skin care company over use of likeness
NBC reportedly holds celebs hostage to Jimmy Fallon's show