Navarro was shown a second yellow card for the infringement after being earlier cautioned for a late challenge.
Antonio Rukavina almost doubled Valladolid's advantage before the interval but his effort whistled past the post.
The hosts' lead was increased to two soon after the break, however, when Perez was again presented with the opportunity from 12 yards following Pape Diop's foul on Omar Ramos.
Levante almost got a goal back in the 73rd minute when Vicente Iborra headed Nabil El Zhar's cross against the bar.
Valladolid become one of only three teams with a maximum six points and lie second behind Barcelona but ahead of Rayo Vallecano.