New York missed a chance to take over the lead in the Eastern Conference, remaining two points behind front-running Montreal.
Sturgis scored in the opening moments of the second period, taking a corner kick and lifting it into the upper right corner of the net from close range. Brown's goal came in the 68th minute with Chris Klute assisting.
The Rapids (7-7-5) jumped past both Seattle and Los Angeles into a tie with Vancouver for fourth place in the Western Conference. New York slipped to 8-7-4.