As of 12:01 a.m. Monday, thousands of cargo containers stacked on ships bound for the U.S. and pallets of Chinese products loaded into the cargo holds of airplanes just got more expensive.

The 10% levy on the declared value of goods U.S. importers buy from Chinese suppliers will add new complications, along with higher costs, as buyers and sellers across trans-Pacific supply chains complete the nuts-and-bolts of how and when the new tariffs are handled.

Customs brokers said they were working overtime in the days leading up to the new tariff deadline, filing entry paperwork for shipments from China and hoping those electronic filings would be processed before the midnight cutoff.

“This is all done electronically and has been for years,” said Marianne Rowden, president of the American Association of Exporters and Importers. Once the tariffs go into effect, the funds are withdrawn directly from the importer’s bank account as soon as the products enter U.S. commerce.

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Customs broker A.N. Deringer Inc. handles the duty payments for many of its customers via electronic transfer and bills the importers later. But it has been encouraging its customers to set up their own transfer accounts with U.S. Customs so the broker doesn’t have to front the funds.

As the U.S.-China trade dispute has heated up in recent weeks, shippers and customs brokers have been scrambling to iron out such details.

Ms. Rowden said shippers were working to get their goods on ships and planes as quickly as possible before the tariffs were imposed, and pushed their customs brokers to file the necessary paperwork. In many cases, if the paperwork has been filed, goods might be considered to be in U.S. commerce even before the vessels they are traveling on have reached U.S. ports, Ms. Rowden said.

But if goods require additional screening by another government agency—for example, seafood products that require the U.S. Department of Agriculture approval to clear the border—that could cause delays and those goods may not have made it in time.

And it isn’t just government screenings that can hold up shipments, said Nina Luu, chief executive of supply chain software company Shippabo. With big ocean carriers forming alliances to consolidate shipments on larger vessels, some importers have had to wait longer to find an available slot on a U.S.-bound ship, possibly missing the tariff deadlines.

“It’s been like, ‘Wow’ for everybody.” Ms. Luu said.

Customs brokers say they already are looking at preparations for the possible increase in the latest U.S. tariffs to 25% on Jan. 1, an increase the White House has threatened if it isn’t happy with the progress of trade talks.

“With something like this coming, most companies that are subject to it are sophisticated enough to be paying attention, and they’re going to make sure their goods get here,” said Mike Lahar, corporate compliance manager at customs broker A.N. Deringer Inc.

Still, there are factors that can’t be planned for.

One Shippabo customer had 50 containers aboard a ship that was delayed by a typhoon in Asia last month. The goods, worth “at least a couple million dollars,” according to Ms. Luu, didn’t arrive in the U.S. until after Aug. 23, when an earlier round of U.S. tariffs went into effect, and the shipper had to pay up.

“There’s very little we can do,” Ms. Luu said. “It’s typhoon season.”

Write to Erica E. Phillips at erica.phillips@wsj.com