Osinbajo spoke during a condolence visit to the family of the late monarch at his Victoria Island Palace.
Oniru died on Monday after a brief illness.
Aged 82, the traditional ruler is survived by wives and children, among whom are a former Lagos State Commissioner for Waterfront, Adesegun Oniru, and an On-Air personality, Tolu Demuren, also known as Toolz.
The vice-president said that the late Oniru’s guidance and counsel all the time was impactful.
“I served in this state as Attorney-General and throughout that period, he was always there for us, always there to support us.
“So, we are all deeply saddened about his departure and we have all lost a father, we have all lost a dear father.
“We thank God that he lived a life of significance; he lived a life of serious contribution to our state and the nation.
“So, on behalf of the Federal Government and President Muhammadu Buhari, I express sincere condolence to the Oniru family and to our state, Lagos State,’’ he said.
Responding on behalf of the family, Adesegun thanked the vice-president for finding time to come to condole with the family.
He recalled that while Osinbajo served as the Attorney General of Lagos State, he also served in that cabinet which had contributed a lot to the nation.
Osinbajo had earlier signed the condolence registrar.