Andres Manuel Lopez Obrador, who has promised to crack down on tax evasion and invest in job-creating infrastructure projects, has a slight lead over Harvard-educated conservative Felip Calderon in opinion polls.
Roberto Madrazo, of the once powerful Institutional Revolutionary Party, or PRI, is lagging behind in third place.
The US is keeping a close eye on the election, hoping for a reversal of the trend that has seen left-leaning governments in power in Brazil, Bolivia and Venezuela.
Soledad Loaeza, a political scientist, said: “The contest has been so heated that it will very likely boost turnout. People recognise it’s an important election and want to participate.”
Of the 71 million Mexicans with voting cards, just over 60% are expected to vote.
Calderon, who wants to cut corporate taxes if elected, has sought to link Lopez Obrador to Hugo Chavez, the Venezuelan leader who is strongly critical of the US.
The former mayor of Mexico City has ridiculed the comparison, and analysts say that he would focus largely on domestic policies.
“He is the only one with a heart, who cares for the people,” said Sanchez Juarez, an 87-year-old Mexico City resident who is supporting Lopez Obrador.
However Marisol Castro, a nutritionist from the western city of Zamora, said Lopez Obrador “has some ideas that seem radical to me. It’s almost as if, if you’re not poor, he doesn’t want to know about you.”
Lopez Obrador insisted at his final election rally on Thursday that he would make good on his promises.
“We will work in a responsible manner, we will not provoke a crisis,” he told a huge crowd in the capital’s historic Zocalo square.
Some say Mexico needs strong leadership at a time when 400,000 Mexicans emigrate each year and drug wars rage.
Porfirio Munoz Ledo, a senior adviser to Lopez Obrador, said: “There are areas of the country that the government doesn’t even control. The drug cartels control them, so we should give thanks if the Mexican government can recover its sovereignty.
“If we can’t do that, we won’t have good relations with anybody abroad.”