Mr Mursi could be sworn in on 30 June, although questions remain over the extent of his authority.
The ruling military council has taken control of many of the president's powers and has dissolved parliament.
"There is no room now for the language of confrontation," he said, after the election authorities declared that he had won 51.73% in the 16-17 June presidential run-off.
Mr Mursi will now have to begin building a civilian administration and the focus will now turn to his choice of prime minister.
The BBC's Jon Leyne in Cairo said there was some talk of him having discussions with Nobel peace-prize winner Mohamed ElBaradei, which would provide reassurance to non-Islamists.
Mr Mursi has also promised to appoint a range of vice presidents and a cabinet of "all the talents".
'Revolution goes on'
On hearing the news of his victory, tens of thousands of Muslim Brotherhood supporters cheered in Cairo's Tahrir Square, chanting, "Down with military rule!".
Celebrations continued until the early hours of Monday morning.
"The revolution goes on, carries on until all the objectives of the revolution are achieved and together we will complete this march" he said.
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