A statement by Laolu Akande, Senior Special Assistant to the President on Media and Publicity on Tuesday in Abuja, said Osinbajo assured Nigerians that anybody involved in the act would be punished.
It was gathered that the sack was announced while Daura was meeting with the Acting President.
It is the first such incident since Nigeria became a democracy in 1999 and coincides with increased tension between the National Assembly and the executive ahead of an election in February 2019 when President Muhammadu Buhari will seek a second term.
Osinbajo had earlier directed Daura to hand over to the most senior director in the agency.
However, Idris was said to have turned down the request, saying he already has enough officers on ground manning the National Assembly.
"That was the explanation, which the IGP also tendered to the Acting President, Yemi Osinbajo, who summoned both Daura and Idris to the Villa to explain their roles in the siege on the NASS".
Security forces temporarily blocked the entrance to Nigeria's parliament on Tuesday, preventing lawmakers and others from entering for up to an hour, witnesses said.
The invasion is believed to be part of plans to impeach the Senate President, Bukola Saraki.
Mr Melaye, while briefing his colleagues at the National Assembly lobby on the impeachment plot by members of the PSG, said the APC national chairman Adams Oshiomhole was in a meeting with Pro-Buhari senators to collect signatures for Saraki and Ekweremadu's impeachment.
The PDP commended federal lawmakers, Nigerians, the media as well as the global community "who rose in defence of our democracy, even as we remind all not to let down their guards".