The Sounders advanced to Monday's Western Conference final against the winner of Thursday's matchup between Sporting Kansas City and Minnesota United.
That winner will claim a berth in the December 12 MLS Cup final against Sunday's Eastern Conference final winner -- either the New England Revolution or Columbus Crew.
After a scoreless first half, the Sounders struck quickly in the second off a corner kick by Uruguayan midfielder Nicolas Lodeiro.
The ball dropped into the heart of the penalty area and O'Neill, a 27-year-old Irishman, leaped high to deflect the ball inside the far post and give Seattle the lead.
It was O'Neill's first MLS goal since 2014 and first playoff marker.
The Sounders, seeking a third title and fourth trip to the final in five seasons, silenced Dallas from there to complete the 10th playoff shutout for Seattle goalkeeper Stefan Frei of Switzerland.