U.S. President Barack Obama called Morsi to congratulate him and pledge to "support Egypt's transition to democracy and stand by the Egyptian people as they fulfill the promise of the revolution."
Earlier, the White House issued a statement calling on Morsi "to take steps at this historic time to advance national unity by reaching out to all parties and constituencies," including respecting the rights of women and religious minorities such as Coptic Christians.
Others described Morsi's election as momentous but stressed that it doesn't mean the county's "revolution" or its problems are over.
Egypt's economy continues to struggle, with widespread poverty, high unemployment and its vaunted tourism sector still sagging on the heels of the political unrest.
With mass demonstrations and clashes with authorities common, the security situation remains tenuous.
Officials warned Sunday -- in advance of the declaration about the presidential winner -- that they were ready to carry out their longstanding policy of using deadly force against people who attack government buildings. More than 1,800 ambulances were dispatched across the county before the announcement in anticipation of election-related violence, the state-run EgyNews agency reported.
And even after Morsi's win, the political situation remains very much unsettled amid lingering questions about whether the military will loosen its grip on power.
Under an interim constitutional declaration, the military council said it retains the power to make laws and budget decisions until a new constitution is written and a new parliament is elected. The declaration said Supreme Council members "shall decide all matters related to military affairs, including the appointment of its leaders." Once sworn in as president, Morsi can declare war but only with "the approval" of the Supreme Council.
Wrote Wael Ghonim, the then-Google executive who helped organize the 2011 revolution, on Twitter: "The first elected civilian Egyptian president in the history of modern Egypt. The revolution continues."