Mr Gayoom, Asia's longest-serving leader, congratulated Mr Nasheed after final results were confirmed.
Mr Nasheed said his victory, in the nation's first democratic presidential poll, showed the people of the Maldives were embracing the future.
President Gayoom won the first round this month, but failed to secure the 50% needed for outright victory.
With all the votes counted in the second round, Mr Nasheed, a former political prisoner, won 54% to Mr Gayoom's 46%.
'Close race'
"I congratulate Anni [Mr Nasheed's popular name]," Mr Gayoom said in a radio address.
"I thank the people of the Maldives for allowing me to serve them for 30 years."
Mr Nasheed told the BBC he had spoken to Mr Gayoom and that the two men would meet later in the day.
The outgoing environment minister said the elections had been a "very close race".
Abdullah Mausoom said the country's first "multiparty elections have been held freely and fairly".
The election was the culmination of reforms in the Indian Ocean islands that followed pro-democracy street protests and international pressure.
Mr Gayoom, 71, has ruled the Maldives uncontested since 1978, elected back into office six times by referendums.
The BBC's Roland Buerk in the capital, Male, said Mr Gayoom's supporters had credited him with overseeing an economic expansion fuelled by tourism.
But Mr Gayoom's critics say he was a dictator who ruled like a sultan of old, says our correspondent.Source: news.bbc.co.uk
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