Police said the incident happened in the vicinity of the Shangri-La hotel where a three-day meeting of top defense officials is taking place. Carter was in the hotel at the time, conference organizers said, but was in no danger and neither were any other delegates.
US officials travelling with the secretary said they were unaware of the incident until hearing news reports several hours later, Reuters reported.
Carter and his delegation left later in the morning for Vietnam.
The police said the car was stopped at a checkpoint but when the driver was asked to open the rear of the vehicle for checks he accelerated away, crashing through barriers.
The police said they opened fire on the vehicle and it came to a stop on a road adjacent to the hotel. Substances believed to be drugs were found on one of the persons detained, the police department said in a statement on its Facebook page.