SINGAPORE—A tariff agreement with Vietnam emphasizes one of the White House’s top priorities in this frantic round of dealmaking over global trade: Slamming shut any back door routes for Chinese goods to enter the U.

With the dust far from having settled on this week’s US-Vietnam trade agreement, one thing is clear: companies operating in what’s become one of the most vibrant Asian economies are on notice to move up the value chain.

The US trade agreement with Vietnam reportedly lowers tariffs but lacks detail, triggering sharp warnings from Chinese officials.