The home side threatened early on as Nick LaBrocca brought the ball under control just outside the penalty area, only to see his powerful volley well saved by Donovan Ricketts, who tipped the ball over the bar.
Portland took the lead just before the break when Diego Valeri fed the ball to Fanendo Adi, and the striker knocked the ball back into the path of the midfielder to rifle an effort through the hands of John Berner.
Dillon Powers took the spot kick and he placed the ball into the bottom corner, sending Ricketts the wrong way.
The Timbers could have retaken the lead immediately as Jorge Villafana raided down the left and lifted a cross into the area for Adi, but the striker mistimed his header and his effort went harmlessly over the bar.
Colorado then had an excellent chance of their own to go ahead when Brown raced onto a long ball over the top of the Portland defence, but he hit his shot straight at Ricketts.
However, Brown atoned for his miss when he puts his side in the lead with a strong header, as he latched onto Marc Burch's cross and placed the ball into the top corner.
Brown had the chance to score his second and seal the game for the home side when he was put clean through on goal, but was once again denied by Ricketts.
This was to cost the Rapids as Portland levelled things up when the ball fell kindly for Gaston Fernandez in the box following a flick from Maximiliano Urruti, and the striker buried a low effort past Berner to earn his side a point.