Combine the oil, margarita mix, lime juice, minced garlic, chili powder, ground cumin, and cayenne pepper in a large zip-top bag and seal; shake to mix.
Add the chicken leg quarters to the bag, seal, and turn the bag several times to coat the chicken evenly with the marinade.
Refrigerate the sealed bag and marinate the chicken for 2 to 8 hours.
Preheat the oven to 350°F (175°C) and lightly grease a 9×13-inch baking dish.
Place the marinated chicken in the baking dish skin-side up and pour any remaining marinade over the pieces. Season the chicken with salt and pepper to taste.
Bake the chicken uncovered for 45 minutes, then switch the oven to broil and broil 2 to 3 minutes until the skin is browned; watch closely to avoid burning.