WASHINGTON—President Donald Trump ran for office promising changes to the H-1B visa program, which brings high-skilled foreign workers to the U.S. But come Monday, the government opens another round of applications using rules that have long been in place.

Demand for the program badly outstrips the supply so, as in past years, the government will use a lottery to decide which companies get them. The visas are coveted by tech firms, who have unsuccessfully lobbied Congress to increase the cap, which is set at 85,000. The new...