Both sides failed to convert penalty chances, but George John scored to give FC a leg up in the Supporters' Shield race.
The game heated up after the hour mark.
LA defender Leonardo was red carded in the 60th minute after his poor defending gave away the ball to Blas Perez in the penalty area, and he brought the Dallas striker down from behind.
His transgression wasn't punished as harshly as it might have been, however. Carlo Cudicini guessed Kenny Cooper's penalty correctly and palmed the ball away to keep the game scoreless.
It was then Dallas' turn to give away a spot kick, and LA's turn to squander it. A spot of contact from George John caused Jose Villarreal to trip with the ball inside the box, before Landon Donovan saw his shot from 12 yards saved by Raul Fernandez.
John atoned for his error in the 87th minute. Eric Hassli's header re-directed a corner kick at the LA crossbar, where it bounced down for John to head in the winner.
But as he was celebrating, John went down to the ground, having been struck by an object from the crowd. Clutching his head and bleeding, he received treatment for several minutes.
The win gives Dallas a 5-1-1 record with 16 points, best in MLS. LA is 2-1-2 with eight points.