The Democratic Party had announced its win earlier on Monday.
"Mongolia is meeting a new morning, with a new president," Altankhuyag Norov, the DP chairman, told a gathering at the party headquarters.
Elbegdorj and his supporters then marched onto the main square in Ulan Bator, where they celebrated at the foot of a statue of Genghis Khan.
Al Jazeera's Hywel Davies, reporting from Ulan Bator, said DP officials were claiming their candidate had won 52-53 per cent of the vote, giving him a very narrow win.
Message of change
The presidential elections were held amid concerns over falling mineral prices and fears of a repeat of the deadly riots that followed last year's parliamentary polls.
|Elbegdorj had centred his campaign on a message of change [EPA]
Elbegdorj, who had pitched his campaign around a Barack Obama-style message of change, has said he will work to ensure Mongolia's natural resources are used "to improve the livelihood of the Mongolian people".
Though the office of president is largely ceremonial in Mongolia, the contest for the presidency has distracted parliament from work on approving a landmark agreement on investment in the country's mining industry.
One deal involving a major copper and gold project is thought to be worth up to $3bn to the poverty-stricken country's economy.
One-third of Mongolia's population lives below the poverty line and unemployment has risen as the global economic crisis has pushed down mineral prices.
Fears of unrest
Alcohol sales were banned on Sunday and sports events, cultural shows and other large gatherings were cancelled in an attempt to a repeat of last year's violence, which left five people dead.
In an attempt to avoid unrest, the General Election Committee urged the media not to report allegations of fraud.
"The media and television are showing comments from ordinary people and there are many different things being said about fraud and cheating, information that is not confirmed," Namsraijaviin Luvsanjav, the head of the committee, said.
Mongolian elections have previously been plagued by claims of fraud and bribery.
In order to prevent electoral fraud, Mongolians were for the first time required to present a special voter card when entering polling booths, as well as their identity documents.