The easiest way to combine study abroad with a work experience is to choose a study abroad program that includes an internship or service-learning experience. See International Internships for more about study-internship programs.
Direct enrollment in a foreign university is a possibility for post-graduate studies. Tuitions for direct enrollment are low in some countries (low, for example, in much of continental Europe and many regions outside of Europe, but higher in popular destinations such as Britain, Ireland, and Australia). Student status may allow you to work part-time or during vacations in some countries. The main disadvantage of attempting to work your way through an overseas university is that you may find yourself marking time, neither making headway towards a degree nor progress in a career. You would also want to be sure that a degree earned from an overseas university would be well-recognized in the U.S., as would be a degree earned from a U.S. university.
Inquire on work abroad positions that qualify as internships or are internships administered by a study abroad program.