Staff member Esther Kidemba died from severe head injuries, Strathmore University confirmed in a statement.
Panic spread after security officials fired gunshots as part of the drill at midday on Monday, local media report.
Several students said they had not been given advance warning of the security exercise.
Most of the injuries were caused by people jumping from windows, a medical source told AFP news agency.
"We were not informed... we just heard gunshots and had to run for our dear lives," one student told the local Capital FM news website.
Strathmore University said in an official statement that the drill had been carried out to test "the preparedness of the university community and emergency team in the event of an attack".
It added that teams of students and staff had been trained in "evacuation, assembly points and exit points", but did not elaborate on whether people had been notified about the timing of the drill.
"Unfortunately some students and staff panicked and got injured," it said in a statement released before the news of Ms Kidemba's death was confirmed.
Many Kenyans have taken to social media to condemn the university over the apparent mishandling of the exercise and for putting the lives of students and staff at risk.
"What happened today at Strathmore University was an attack, not a drill!," commented one twitter user.
The university has said it will cover the medical costs of those injured as a result of the exercise.
Nairobi police chief Japheth Koome said all the proper procedures were followed for Monday's drill, Reuters news agency reports.
In April. the Kenyan government instructed the country's higher education institutions to provide training to teach students how to respond in the event of a terror attack.
It followed the killing of at least 148 people, mainly students, in an attack by Islamist militant group al-Shabab on Garissa University College in north-eastern Kenya.
One woman has died and at least 31 others have been hospitalised after a security exercise at a university in the Kenyan capital Nairobi.