The hosts surged ahead in the 10th minute when David Texeira headed home from close range.
Visitors Sounders restored parity midway through the first half when Clint Dempsey found the target with a right-footed free kick.
However, Dallas regained their advantage three minutes before the break.
Sounders, though, launched a comeback in the second half and picked up all three points.
An own goal from Stephen Keel on 75 minutes saw Sigi Schmid's side make it 2-2, and then with five minutes of normal time remaining, Dempsey turned in the winner from close range after latching onto Chad Barrett's pass.