The game looked destined for a goalless draw before some late injury-time drama.
Daniel James' cross inside the Palace box was followed with several claims for a handball from the Leeds players.
Referee Kevin Friend consulted the pitchside monitor and eventually pointed to the spot after it was clear the ball had struck the arm of Eagles stopper Marc Guehi.
Brazilian star Raphinha stuttered up to the spot and slotted the ball to the left of Palace goalkeeper Vicente Guaita.
Elland Road was bouncing after the goal as Leeds sealed a valuable win which moves up into 15th and five points above the drop zone.
Speaking after the game, Vieira said he didn't want to focus on the penalty decision.
"I'm frustrated because we were the better team. When you don't put away your chances, you know you can get punished," Vieira said.
"I don't want to focus on the penalty. We had a couple of good chances, especially in the second half. We didn't have the technical ability to play the right ball at the right time in the right space. I thought we were really unlucky.
"It's a difficult place to come. We knew the chances would come in the second half. If we defend that way away from home, we'll win more matches than we lose.
"The last game against Villa we didn't perform at all, but in this one we performed. We just have to accept the result. Some of the games have gone in our favour – tonight it didn't.
"We don't have time to feel sorry for ourselves. We just have to focus on our next game."
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