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 controversial foul on Nat Borchers inside the penalty area at the end of a long run from midfield. Replays showed that Borchers hit the ball first.
"It just doesn't seem like the soccer gods are with us right now," said Borchers. "It's unfortunate for us to go behind like that. I'm not sure if it was a penalty or not. I didn't think it was. But I thought we fought back. We had a lot of energy after we gave up the goal."
"It's really tough," added defender Chris Wingert. "I don't know what to say. It was a great goal by Plata, a good team goal with great passes. We created a lot of chances. We just weren't able to bury any more than the one."
RSL next hosts Montreal July 24th at 8:00 on CW-30.