The 72-year-old former military ruler also pledged reconciliation with political opponents who fear a return to his autocratic regime of the 1980s, and a government representing "all Nigerians".
"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," Buhari said in an acceptance speech after his dramatic win.
"We shall spare no effort until we defeat terrorism."
He also promised to address a culture of political impunity and the "evil" of rampant corruption but said there would be no witch-hunt and his beaten opponent Goodluck Jonathan had "nothing to fear".
"Our long night has passed and the daylight of new democratic government has broken across the land. 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."
UN chief Ban Ki-moon and US President Barack Obama led the congratulations for Buhari.
But it was the first unprecedented opposition victory against a sitting president in Nigeria that won most plaudits.
"The last few days have shown the world the strength of Nigeria's commitment to democratic principles," Obama said, while his top Africa diplomat said Nigeria had raised the bar for the continent.
"Nigeria is a trailblazer," Assistant Secretary of State for African Affairs Linda Thomas-Greenfield told AFP.
Buhari had praised Jonathan for his statesmanship in conceding defeat before the final result was declared, helping to defuse the potential for poll-related violence and lengthy legal wranglings.
In a statement, the heads of international election observation missions, including the African Union and the European Union, congratulated Nigeria and its people "for their commitment to a peaceful response to the election results.
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