Kenny Cooper eventually put the hosts in front midway through the first half when he collected Obafemi Martins's pass and shot home from inside the Crew box.
Federico Higuain was unlucky not to restore parity for Gregg Berhalter's side two minutes before the break when his effort from close range came back off the crossbar.
The Sounders conceded the equaliser just before the hour mark. Djimi Traore brought down Oduro inside the box and was sent off for his challenge, resulting in Higuain successfully slamming home the spot kick.
The numerical advantage allowed Crew to push forward for a winning goal, and they eventually got it in the fourth minute of injury time when Justin Meram turned in Higuain's corner from close range.