"I promised the country free and fair elections. I have kept my word," Mr Jonathan said in a statement.
He said he had conveyed his "best wishes" to Mr Buhari, and urged "those who may feel aggrieved to follow due process... in seeking redress".
A spokesman for Gen Buhari's All Progressives Congress (APC) party praised Mr Jonathan, saying: "He will remain a hero for this move. The tension will go down dramatically."
Gen Buhari's supporters took to the streets in APC strongholds, including the northern cities of Kano and Kaduna, to sing and dance in celebration.
Analysis: Will Ross, BBC Nigeria correspondent, Abuja
Gen Buhari's victory is a hugely significant moment in Nigeria's turbulent history. Never before has a sitting president been defeated in an election.
Since independence from Britain in 1960, there have been numerous coups and most elections have been rigged. Of course in a close election there will be many voters who are not pleased with this outcome but the whole process is a sign that democracy is deepening in Nigeria.
The poll has once again brought to the surface dangerous religious and regional differences and there is still a threat of violence.
The man who has been voted out, Goodluck Jonathan, has played a huge part today in trying to prevent that. He made the phone call when there would no doubt have been some in his camp who would have preferred to dig their heels in.