BRUSSELS — The European Union on Wednesday ordered the Dutch government to recover money from Starbucks and told Luxembourg to claw back funds from a Fiat Chrysler unit, in an expanding crackdown on tax avoidance by corporations.

Margrethe Vestager, the antitrust chief of the European Union, said that Luxembourg and the Netherlands had given the multinational corporations illegal state aid by letting them shift profits and pay lower tax rates than those available to other companies.

The decisions are a sign of Europe’s determination to counter increasingly sophisticated tax strategies used by multinational companies. And Wednesday’s moves could be only the first of a series of enforcement actions by Ms. Vestager’s office, which has been investigating tax arrangements that some European countries have used to attract multinationals, including big American technology companies like Apple in Ireland and Amazon in Luxembourg.

Other European Union members say the tax deals often amount to unfair competition and siphon tax revenues from bigger markets where the companies might actually do more business. And American officials have raised questions about whether some multinationals are using European tax shelters to avoid paying their full share of taxes in the United States.