Witnesses said the blasts sent up a cloud of smoke and dust near the campus, the scene of repeated clashes in the past few months between Islamist students and police.
The third bomb struck close to the main gates, where police investigators and journalists had gathered, causing no casualties.
The bombings were the latest in a spate of attacks against the security forces since the army overthrew elected Islamist president Mohammed Morsi last July.
A fourth bomb placed in a car parked near the university was defused, security officials and state television said.
The interior ministry identified the slain officer as Brigadier General Tarek al-Mergawi.
An assistant interior minister, Major General Abdel Raouf al-Serafi, and four other policemen were wounded.
"I was waiting for the bus when I heard two explosions. There was dust in the air and policemen were screaming," said a witness, Sakta Mostafa.
A Cairo University student said he ran out of the campus after hearing the blasts.
"I found a lifeless man in plain clothes and a policeman bleeding from his leg," said the student, Amr Adel.
A senior detective, Mergawi would have been in civilian clothes.
Amateur footage posted on an Egyptian newspaper's website showed policemen running out from a cloud of smoke and dust sent up by the first explosion.
The second bomb went off moments later.
Interim prime minister Ibrahim Mahlab led Mergawi's funeral procession, as policemen carried the coffin draped in a red shroud.
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