Vancouver had recovered from a 1-0 deficit at half-time to equalise through Erik Hurtado in the 64th minute but when the referee judged that Whitecaps defender Kendall Waston had handled the ball in the box, Dallas' chance to win had arrived.
Up stepped Brazilian full-back Michel and he converted from the spot to set up a semi-final with Supporters' Shield winners Seattle Sounders.
The Whitecaps pushed for a second equaliser in the final six minutes plus stoppage time but were unable to breach the home side's defence again, with the Canadian club failing to reach the final four of their conference - a stage they have never reached in four seasons in the MLS.
Vancouver felt hard done by with the penalty decision, as the ball bounced up and struck Waston on his right arm, which was not raised.
After the final whistle, Whitecaps midfielder Sebastian Fernandez confronted the referee, launching an abusive tirade, which earned him a red card.
Dallas opened the scoring with five minutes left in the first half - Tesho Akindele striking back across Vancouver goalkeeper David Ousted after being released by Mauro Diaz.
Hurtado's equaliser came after the hosts failed to properly clear a Whitecaps corner, with the striker notching his first goal since September 12, when he had also scored against Dallas.
But with Michel scoring his ninth goal of the season, Dallas advanced.