The embassy "wishes to advise American citizens of the potential danger that exists when moving about the city during this time and to exercise extreme caution should travel become necessary."
The statement also asked American citizens to obey a curfew issued in Khartoum State from 6.00 pm to 6.00 am.
The British Foreign Office in London also advised its nationals against all but absolutely essential trips to Sudan "until further notice".
Although the UN representative in Khartoum has said the crash that killed Garang was an accident, with no suggestion of foul play, Sudanese President Omar al-Beshir announced he will launch an investigation.
Over 100 people have been killed in three days of rioting in the Sudanese capital and the flashpoint southern town of Juba, officials said.