Rabbani, chairman of Karzai's High Peace Council, was killed by a turban bomber who had purported to be a peace emissary from the Taliban leadership.
His death threatens fresh turmoil in Afghanistan, with experts saying it could lead his Tajik supporters, who provided the bulk of resistance to the Taliban when they were in power to reject all efforts to talk peace.
Officials also revealed that last week they arrested a man who spoke to the killer on the phone the day before Rabbani's killing.
Police and other senior figures have blamed the Taliban for the attack, but unusually, the leaders of Afghanistan's ten-year insurgency have so far refused to comment on it.
The investigation will be led by Afghan defence minister Abdul Rahim Wardak, seen as close to Karzai, a statement from the presidential palace said.
The statement added: "As a result of actions taken to identify the elements behind Ustad (professor) Rabbani's assassination, an arrest has been made and information obtained from him is bringing us closer to the truth."
A government source speaking on condition of anonymity told AFP that the arrested man had spoken to the bomber the day before the attack.
The bomber, who was believed to be a trusted Taliban emissary, had left the official guesthouse where he was staying to speak to the man, the source said.
"When Esmatullah (the bomber) left the guesthouse he had a white hat but when he returned, he was wearing a turban and had new clothes. He had cleaned himself up," the source said.
"He joked with the workers of the guesthouse about his new clothes and turban, saying: 'I'm going for a very important meeting. I should look good.'"
The statement from the presidential palace also announced beefed-up security for senior figures in Afghanistan in the wake of Rabbani's killing.
Rabbani was buried in Kabul on Friday.
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Afghanstan should involve UN to investigate. Those reponsible and their master should be bought before the ICC.