Strathmore University said in a statement that one staff member had died as a result of injuries received during the drill aimed at testing the preparedness of the university community and emergency response team in the event of an attack.
In a statement, the Nairobi institution identified the woman who died as 33-year-old Esther Kidambi, a staff member working in its catering department.
Ms Kidambi died from head injuries after jumping from a third floor window, according to the local Star newspaper.
Others suffered broken legs, including a professor who jumped from the fourth floor.
Some students told local radio station Capital FM that loud bangs and people pretending to be extremist attackers caused panic at the university, causing everyone to flee to the exits.
Memories of the April 2 attack by four gunmen from Somali extremist group al-Shabab that killed 147 people are still fresh with many Kenyans.
Kenya has experienced a wave of bomb and grenade attacks since it sent its troops to Somalia to fight al-Shabab in October 2011.