The fugitives crawled through the pipe, which leads from the prison to a main street, on Thursday night, said police Lt. Gen. Mohammed Abdul-Rahim Al-Tayeb.
"The federal police are investigating the matter and searching for the fugitives," Al-Tayeb said in a statement carried by the state news agency.
Al-Tayeb said the fugitives exchanged fire with security personnel manning a checkpoint on the outskirts of the capital. In their pursuit, police arrested the driver of a car transporting the convicts, but the four men were not recaptured, he said.
The men were found guilty of killing John Granville, from Buffalo, New York, and his driver in a drive-by shooting as the 33-year-old diplomat was returning from a New Year's Eve party in Sudan's capital on Jan. 1, 2008.
Granville worked for the US Agency for International Development and was helping implement a 2005 peace agreement that ended more than two decades of civil war between south and north Sudan.
In Washington, the State Department said it expected Sudanese authorities to apprehend the escaped convicts and ensure that justice is served.
Granville was the first American to be killed in Sudan since 1973, when two diplomats were slain by Palestinian militants.
During the trial, the prosecutor said the assailants acted out of religious zeal. A Sudanese investigator said the men were looking for a Western target and had also plotted a failed attack on a British diplomat.
One of the convicts was an active duty army officer at the time of the slaying.
