Los Blancos led fierce rivals Barcelona by 10 points at one stage but now find their advantage down to just six after recording consecutive draws.
The Portuguese forward played the ball into Ozil's feet and then received the German's back-heeled return before rounding Villarreal keeper Diego Lopez and slotting into an empty net.
With Jose Mourinho's side seemingly headed towards victory, the Yellow Submarines grabbed an unlikely equaliser eight minutes from time.
Marcos Senna's free kick went straight through the Madrid wall and past Iker Casillas into the visitors net.
A remarkable spell of three red cards then followed as Mourinho was sent to the stands for sarcastically applauding the referee.
Ozil and Sergio Ramos joined their manager in seeing red as Madrid finished the match with nine men.