Elbegdorj, regarded as a liberal and populist, 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.
"I think we should have riots if Enkhbayar wins"
a 32-year-old voter
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.
Alcohol sales were banned on Sunday and sports events, cultural shows and other large gatherings were cancelled in an attempt to prevent last year's violence that left five people dead from happening again.
"I hope there won't be riots this time," Garda, a 75-year-old retiree who voted for Enkhbayar, told the AFP news agency.
"He did a great job in the past four years and he needs time to finish what he started," he said.
'Fraud and cheating'
But for Tseren, a 32-year-old businesswoman, unrest was a strong possibility.
"I think we should have riots if Enkhbayar wins because he plays a dirty game," she said before voting for Elbegdorj.
"I hate Elbegdorj, but we have to have change."
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 an attempt 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.