Both sides struggled to create chances in a nervy start to the game, before Dallas goalkeeper Chris Seitz was forced into action midway through the first half to keep out a powerful long-range effort from Pedro Morales.
Vancouver appeared to be taking control of the match until they saw the deadlock broken by the hosts, who took the lead in the 40th minute when Tesho Akindele collected a clever pass from Mauro Diaz before firing home.
The half-time interval helped Vancouver produce a response early in the second half, but their frustration continued as Mauro Rosales struck the crossbar with a free kick.
However, an extended spell of pressure was finally rewarded just after the hour mark when Erik Hurtado took advantage of the home side's inability to deal with a free kick into the box by drilling in the equaliser from close range.
Vancouver came agonisingly close to pulling ahead moments later, but Kekuta Manneh could only drag a shot just wide of the post from a tight angle.
It was a Kendall Watson mistake which gifted Dallas the chance to book their spot in the next round as the referee decided that the defender had handled the ball inside the box.
Michel took on the responsibility to slot home the penalty and ensure it will be Dallas who meet the Seattle Sounders in the Western Conference semi-finals.