WASHINGTON—The U.S. sanctioned a multibillion-dollar network of Iranian companies, banks and funds accused of financing the country’s elite paramilitary unit, ratcheting up global pressure on Tehran and sending a warning to governments and companies considering continued engagement with Iran.

By targeting the Basij militia’s financing network and citing the group’s alleged use of child soldiers and other human-rights abuses, the U.S. hopes to not only choke off funding to the prominent Islamic Revolutionary Guard Corps unit, but also scare off any business dealings with the country.

“The IRGC is pervasive within the Iranian economy,” a senior administration official said. “This is precisely the kind of activity that we have warned other companies and governments about extensively.”

Many firms are pulling out of Iran as the U.S. rolls out an escalating and economy-crippling sanctions campaign meant to force Tehran to negotiate a new nuclear and security deal that addresses an array of U.S. concerns.

But from China to Europe, some governments and companies are considering maintaining financial and trade ties with Tehran as a way to keep the country’s critical oil supplies flowing and to oppose Washington’s decision this year to pull out of the 2015 Iran nuclear deal.