Tax cuts are exactly that—legalization of tax evasion. Some people evade taxes because they are criminals; many more evade taxes because taxes are too high. Lowering tax rates reduces the population of that latter category, freeing up enforcement resources to focus on the former category. Tax revenues and compliance with tax laws go up.
Similarly, some employers violate immigration law because they are criminals, but most violate the law because the law is too onerous. Reduce the law's burden and resources are freed to focus on those with criminal intent. When employers violate unworkable OSHA or EPA or IRS regulations, we generally take their side here. What makes you think the INS (or whatever they call it these days) is the one efficient, benevolent bureaucracy in the gubmint?
Actually, the tax evasion analogy helps show why this will work. The Reagan amnesty didn't structually change immigration law, it just gave some people a free pass. In that sense the Reagan amnesty was like a one-time tax rebate (like those silly "muffler" checks that got mailed out in 2002). It didn't change anyone's long-term behavior because it didn't alter the law's burden. This plan, by contrast, makes a genuine change to the law, and will change behavior the same way tax cuts do.