Alphonso Davies and Fredy Montero scored in Thursday's second leg as the Whitecaps won the all-MLS quarterfinal 3-1 on aggregate.
Carl Robinson's Vancouver will face Mexican outfit Tigres in the final four.
The Whitecaps and Red Bulls had played out a first-leg draw at New York last week to leave the contest in the balance.
But Davies quickly swung the tie in Vancouver's favor with a goal in the fifth minute at BC Place, keeping his composure to place the ball beyond Red Bulls goalkeeper Luis Robles inside the 6-yard box.
Montero — on loan from Chinese Super League side Tianjin Teda — then came off the bench to make his debut and the Colombian's impact was immediate.
After replacing Marcel de Jong in the 67th minute, Montero finished emphatically from a corner nine minutes later to seal the victory.