"As president of the Federal Republic of Nigeria I will discharge my duties and perform my functions honestly to the best of my ability, faithfully and in accordance with the constitution," Tinubu said, in a live broadcast from the capital Abuja's Eagle Square.
Tinubu, a 71-year-old southerner, takes over from the 80-year-old northerner Buhari as Africa's most populous nation faces a sea of economic troubles and a security crisis.
Foreign leaders and representatives present at the ceremony included presidents Cyril Ramaphosa of South Africa, Paul Kagame of Rwanda and Nana Akufo-Addo of Ghana as well as delegations from the US, Britain and China.
Kashim Shettima was sworn in as vice president, taking over from Yemi Osinbajo.
The ruling party duo were declared winners of the February 25 election, gaining 8.8 million votes and picking up the required number of ballots across two-thirds of Nigeria's states.
The main opposition leader Atiku Abubakar, who came second, and outsider Peter Obi, who was third, are contesting the results in court, claiming fraud.
The electoral commission has acknowledged "glitches" during the vote but dismissed claims that the process was not free and fair.
Dubbed a political "godfather", Tinubu campaigned saying "it's my turn" to govern Nigeria, Africa's biggest economy.
He touted his experience as Lagos governor from 1999 to 2007 - a period that, supporters say, modernized Lagos, Nigeria's business hub.
But the new president also faces corruption allegations, which he denies, and questions over his health.