A man, named as Ahmad Mansour, has been identified as the assailant and reportedly charged with murder.
Mr Mansour worked as a clerk at the building where the incident occurred, which oversees payments to teachers at the university.
It appears he became angry when he was unable to pay back a loan of 18 million Lebanese pounds ($12,000) and was forced to sell his car, Education Minister Abdel Rahim Mrad said.
Colleagues dead
Mr Mansour, who had worked at the fund for at least 10 years, is said to have arrived at work with a suitcase containing a Kalashnikov and two pistols.
He then made his way through two offices, shooting colleagues - including the manager - as he went, witnesses said.
Mr Mansour lived in the village of Loubieh, south Lebanon, with his wife and four children.
His wife, Mona Khalil, is reported to have broken down on hearing the news.
"This is a catastrophe... I can't believe Ahmed would do something like this," she said.
She said Mr Mansour was a diabetic and used tranquilisers.
Police arrived promptly at the scene, where they arrested Mr Mansour - who reportedly lit a cigarette after carrying out the killings.
A dozen ambulances were also rushed to the scene, and distraught employees and relatives gathered outside the building to await news of friends and colleagues.
Burdened by debt
Among the dead was Rachel Saade, the fund's manager.
Her husband, Jean-Claude, wept as her body was taken away.
"Poor Rachel, they killed her," he said.
The BBC's Kim Ghattas in Beirut says such incidents are rare, despite Beirut's reputation as a city of violence.
But she adds that the city is burdened by a $30bn debt at present, and living conditions are difficult for many.
This may explain why Mr Mansour got so upset when his car was confiscated over his failure to pay back the money.