The US uses a couple tools to make US drug policy essentially world drug policy. Mainly trade agreements, military agreements(weapon sales and defense agreements), and economic aid.

We attach riders to these agreements which make adoption of US drug law a condition of the deal.

This is in first world countries. In the second and third world countries (the middle east and south america, and to a lesser but not negligible extent Africa) we just enforce our laws by force with or without permission.