Plata received the ball out wide from Kyle Beckerman, took one touch to get inside the penalty area, and sent a blast to the far post.
Real Salt Lake (7-4-8) has won just once in eight matches, but kept Vancouver (6-4-9) from getting its first win in Utah in six attempts.
Robbie Findley ran onto a long ball by Beckerman in the 93rd minute, but his flick inside the penalty area went off the face of Vancouver goalkeeper David Ousted and sailed just wide to preserve the tie.
The Whitecaps took a 1-0 lead in the 73rd minute when Darren Mattocks converted a penalty kick. He drew a foul on Nat Borchers inside the penalty area at the end of a long run from midfield. It was his second consecutive game with a goal.