Dallas led twice in the first half at home but an own goal to Stephen Keel dragged Seattle back to 2-2 with 15 minutes to go before Dempsey finished sharply from close range in the 85th minute.
The win took the Sounders to 10 points from six matches to put them third in the Western Conference, just three points adrift of leaders Dallas.
Dallas had produced the best start in their history in their opening five games and began the game brightly before David Texeira opened the scoring the hosts in the 10th minute.
The Uruguayan striker got hold of a bouncing ball, which went over Seattle goalkeeper Stefan Frei, and nodded the ball into an empty net.
Dempsey, who had scored a hat-trick in Seattle's 4-4 draw at arch-rivals Portland Timbers last week, equalised for the visitors in the 22nd minute with a thumping free-kick from 25 yards, although Dallas led 2-1 at half-time thanks to a penalty from Michel three minutes before the break.
Seattle started to get on top in the second half but needed a slice of fortune to equalise in the 75th minute with Keel providing the decisive touch to Dempsey's wayward shot to deflect the ball past Dallas goalkeeper Chris Seitz.
With five minutes left in regulation time, Dempsey struck the winner for the Sounders with the United States international playing a sharp one-two with Chad Barrett before finishing from the edge of the six-yard box.
Seattle sit just behind second-placed Colorado Rapids on goal difference in the West after the latter won 1-0 at Toronto FC.
Edson Buddle scored the winner for Colorado, who also have a game in hand on Seattle and fourth-placed Real Salt Lake, who moved up to 10 points after a 2-2 draw at Philadelphia Union.
Portland Timbers poor start to 2014 continued as they were pegged back by Chivas USA in a 1-1 draw.
Captain Will Johnson gave the Timbers the lead in the seventh minute in Portland with a powerful strike from the edge of the box but the visitors claimed a point after Erik Torres' tap-in with 11 minutes remaining.
Portland remain winless from six games this season with four draws.
In other results, Robbie Keane scored the only goal of the game as Los Angeles Galaxy defeated Vancouver Whitecaps 1-0, DC United trumped New York Red Bulls by the same scoreline, Chicago Fire drew 1-1 at Montreal Impact and New England Revolution won 2-0 over Houston Dynamo.
