Elbegdorj, regarded as a liberal and populist opposition candidate, 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 for the poverty-stricken country's economy.
The latest surveys put the candidates in a statistical tie, though some showed Enkhbayar with a slight edge.
However, there is concern that a closely fought presidential election could increase the risk of post-election violence and allegations of election fraud.
Disputed parliamentary elections last year sparked riots in the streets of the capital, Ulan Bator, in which five people were killed.
Polling stations are due to close at 1400 GMT.