The visitors, who secured a 3-1 win at home in the first leg of the tie, almost opened the scoring inside the first five minutes when Fanendo Adi unleashed a long-range shot, but it was blocked in the penalty box.
The Timbers had the majority of the possession at the start of the second half, and broke the deadlock within 10 minutes of the restart when Diego Valeri played the ball in for Adi, leaving the striker to slice the ball into the top left corner.
Valeri made another run forward for Portland a couple of minutes later, picking out Asprilla with a cross, but his 20-yard drive did not trouble Jesse Gonzalez in the Dallas goal.
Despite Portland's dominance, Dallas were back on level terms with just over 20 minutes remaining as Mauro Diaz squared the ball to Hollingshead, who smashed it home from the centre of the box.
Diaz was involved again five minutes later, whipping a free kick into the box to set up Blas Perez for a header at the near post, which he nodded down into the bottom corner to put Dallas ahead for the first time, although Portland still led on aggregate.
The home side pressed hard for a third in the final 10 minutes, but Portland managed to clear a corner, before Jack Jewsbury blocked the ball back in.
However, Portland ended Dallas's hopes of a late equaliser as Valeri provided the through-ball for Lucas Melano to guide in from close range just moments after coming off the bench.
Portland will face the winner of the Eastern Conference final between Columbus Crew and New York Red Bulls for the MLS Cup next month.