He won 51.73% of the vote in last-week's run-off, beating former Prime Minister Ahmed Shafiq.
World leaders have congratulated Mr Mursi. The White House urged Egypt to be "a pillar of regional peace".
Days of tension led up to the declaration of the result after Egypt's ruling military council gave itself sweeping powers.
When the winner was announced, tens of thousands of Muslim Brotherhood supporters holding a vigil in Cairo's Tahrir Square erupted in jubilation.
"Down with military rule" they chanted amid wild cheering and explosions of fireworks.
Celebrations continued long into the night, with many of Mr Mursi's supporters still in Tahrir Square on Monday morning.
In a televised address, Mr Mursi paid tribute to the protesters killed in last year's uprising against former President Hosni Mubarak.
He said without the "blood of the martyrs" he would not have been elected.
"The revolution goes on, carries on until all the objectives of the revolution are achieved and together we will complete this march. The people have been patient long enough," he said.