"I congratulate His Excellency Bola Ahmed Tinubu on his victory. Elected by the people, he is the best person for the job," Buhari said on Twitter.
"I shall now work with him and his team to ensure an orderly handover of power."
Buhari urged aggrieved opposition parties which had called for the ballot to be cancelled to be calm and maintain peace despite their misgivings.
"There is no doubt the people's decision has been rendered in the results," he said.
Tinubu, candidate of Buhari's ruling All Progressives Congress (APC), won 8.8 million votes against 6.9 million for opposition Peoples Democratic Party (PDP) candidate Atiku Abubakar and 6.1 million for the Labour Party's Peter Obi in Saturday's election, according to final results.
The Independent National Electoral Commission (INEC) confirmed Tinubu as president-elect after he also secured the required 25 percent of votes in two-thirds of Nigeria's 36 states and capital.
Tinubu will succeed Buhari, a former general, who will step down on May 29 after serving the maximum two terms allowed by the constitution.
But Buhari also noted the challenges placed against the vote's outcome.
"That is not to say the exercise was without fault. For instance, there were technical problems with electronic transmission of the results. Of course, there will be areas that need work to bring further transparency and credibility to the voting procedure."
"None of the issues registered represent a challenge to the freeness and fairness of the elections," Buhari declared.
"If any candidate believes they can prove the fraud they claim is committed against them, then bring forward the evidence."