"I'm pushing ahead with the goal of making Mongolia a country ruled by law. The rule of law will create a responsible and just government," the president told supporters at a rally on Tuesday in a suburb of the capital, Ulan Bator.
"With this kind of government, Mongolia will develop further. I will work with you in order to implement this goal."
|Both candidates say Mongolians must see the benefit of the country's rich resources [EPA]
For his part 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".
A two-time former prime minister, he has campaigned on an Obama-like theme of change and anti-corruption.
Speaking to supporters in the capital's main square earlier this week he said his goals were: "To create much needed jobs, to make Mongolians the masters of Mongolia, to give freedom to our people, to create justice in society, to give Mongolians the right to control our natural resources and wealth."
While the office of president is largely ceremonial, 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 in particular, involving a major copper and gold project, is thought to be worth up to $3bn for the poverty-stricken country's economy.
A strong turnout in the countryside in Sunday's vote may give Enkhbayar a slight advantage, as he has been traditionally supported by the conservative and rural poor.
However, there is concern that a closely fought presidential election could increase the risk of post-election violence and allegations of election fraud.