The Jamaican international scored twice in the first half to put away the visitors early in the game.
The home side was on the board early. Marvell Wynne beat his man down the right flank and whipped in a cross, which Brown headed down and in from six yards out.
Brown had his second before the half. Dillon Powers clipped a ball behind the Dynamo backline, and the Jamaican raced onto the end of it and buried a low shot into the far post.
The Rapids made it three just after the break, thanks to an audacious bit of skill from Kamani Hill. Running to the near post off a corner kick, Hill jumped in the air and flicked the ball off the side of his foot and into the far post.