I think you'll find the correlation is not with a higher minimum wage, but with the total cost per employee as imposed by the state. Frex, in California, wages are only about 30% of the mandated cost of each employee. The other 70% is payroll taxes, workman's comp, insurance, and the like.
That 70% is what an illegal labor force saves you, a cost far more significant than any minimum wage increase.
In fact, in some areas illegal laborers make more than minimum wage, because having ducked out of that 70%, the employer can afford to pay more than they would otherwise, and do so to attract a better grade of illegal worker. The illegal worker thereby earns significantly more take-home pay than he could earn as a "legal" worker. (This is common in the construction trade, for instance.) Not only that, but the illegal worker takes home his entire paycheck, minus no deductions or taxes.