Billions of dollars worth of China-made goods subject to tariffs by the Trump administration in its trade fight with Beijing are dodging the China levies by entering the U.S. via other countries in Asia, especially Vietnam, according to trade data and overseas officials.

The Trump administration has for more than a year sought to weed out the practice known as transshipment, in which Chinese exports typically are minimally processed or altered during a brief stop in a third port and then re-exported as a product originating...