The home side entered the match in fantastic form having won four games on the bounce, but things started poorly for the Sounders as Brad Evans took down Blas Perez in the penalty area in the 14th minute, resulting in an early penalty kick.
Michel stepped to the spot and calmly dispatched the opportunity to give the visitors the lead.
Seattle finally drew level in the 62nd minute when Clint Dempsey fed the ball out wide to Obafemi Martins on the right flank. Martins looked up and delivered a perfect ball to the near post where Lamar Neagle was able to tap it home.
Seattle's confidence was flowing after the goal and the hosts created several quality opportunities and forced FC Dallas goalkeeper Raul Fernandez into several outstanding saves.
The Sounders, as they have done so many times this season, kept pushing and found the winning tally two minutes from time as Martins broke through a tackle on the left wing and sent a low cross into the area where an unmarked Cooper was there to drive it into the net.
The Sounders have now scored seven goals in the final 10 minutes of matches and improve to 22 points from 10 matches, five clear of Real Salt Lake at the top of the Western Conference.
FC Dallas, meanwhile, remain in third place in the West on 16 points.