The Impact’s Calum Mallace tripped Plata as he was attempting a shot in the penalty area and Movsisyan converted with a hard shot from the penalty spot for his sixth goal in 10 games.
"We took a good point on the road," said Impact coach Mauro Biello.
"We had opportunities to increase our lead, but this is a difficult place to play. The guys fought for 90 minutes. They gave everything. Conceding the equalizer is disappointing, but we keep going."
Harry Shipp scored in the eight minute on a left-footed putback. RSL goalkeeper Nick Rimando knocked down Laurent Ciman’s knuckling free kick and Shipp raced in to finish.
"It was nice to get a goal," said Shipp. "We played well enough in the first half to get two or three goals. It would have been nice.
"We worked hard, with the heat and the altitude. Everyone out there today put in a great effort."
Real Salt Lake had two shots in the final minute, but Montreal goalkeeper Evan Bush preserved the draw with a diving save on Kyle Beckerman’s deflected shot.
The Impact (6-4-7) improved their unbeaten streak to five games in their only 2016 regular-season meeting with RSL (8-5-5). Each team ended the night in third place in their respective conferences.
"We could have been up 2-0 or 3-0 and killed the game," said Montreal’s Dominic Oduro. "We played against a very good team. RSL put a lot of pressure on us in the second half. Away from home, a point is good."