"Let me state clearly: President Jonathan has nothing to fear from me," he said in Abuja after his victory was confirmed in the early hours. "He is a great Nigerian and still our president."
He added: "Democracy and the rule of law will be established in the land. Let's put the past behind us, especially the recent past. We must forget our old battles and past grievances and forge ahead."
Buhari on Wednesday warned Boko Haram of a concerted fight-back to defeat the militants after their six-year rampage in the country's northeast.
"I assure you that Boko Haram will soon know the strength of our collective will and commitment to rid this nation of terror and bring back peace," the 72-year-old said in the capital Abuja.
"We shall spare no effort until we defeat terrorism," he said at a ceremony after he was confirmed winner of last weekend's presidential election.