Former Sporting CP star Montero converted a 61st-minute spot-kick as the Whitecaps left Red Bull Arena with a point.
The Red Bulls and Whitecaps rotated heavily midweek, with both teams scheduled to play again on Sunday (AEST).
Vancouver – without a win in three matches – opened the scoring via Scott Sutter, who found the back of the net from an acute angle in the 29th minute.
The Red Bulls, however, equalised eight minutes later thanks to Brian White before an Andy Rose own goal gifted the hosts the lead 10 minutes into the second half.
The visiting Whitecaps earned a point when VAR ruled a handball on Sean Nealis, with Montero making no mistake from the spot.