The visitors stormed into the lead in the third minute when Krisztian Nemeth was played through on goal and the forward unleashed a powerful strike from outside the box into the bottom corner.
The Dynamo edged ahead just before the break as Bruin rose highest in the penalty to thump a header past Marin from Brad Davis's corner.
In the second half, the home side got their third goal as Barnes picked up the ball and unleashed a strike from 25 yards which found the bottom corner of the net.
Kansas City got back into the match when Dominic Dwyer pounced on a mistake in the Houston defence, latching onto an underhit back-pass before rounding Tyler Deric to slot the ball home into an empty net.
The visitors were handed the chance to get an equaliser when Nemeth coasted into the box and fired at goal, only to see his effort blocked by the hand of Jermaine Taylor. The referee pointed to the spot and sent off Taylor before Feilhaber beat Deric from 12 yards to level the scores.
However, there was another twist in the tale when Houston retook the lead as Davis whipped a pinpoint cross towards the near post, where Raul Rodriguez guided a header past Marin.
Kansas City piled men forward in stoppage time and they earned a point as Feilhaber rifled a volley which sailed past Deric into the top corner.