"According to our data we have won the vote by at least 500,000 votes. This is irreversible," leftist Andres Manuel Lopez Obrador told supporters shortly after electoral authorities said they would announce results later in the week.
Minutes later, conservative Felipe Calderon, of the ruling party made a similar claim. "We have won the presidential elections," he said, pointing to several projections that put him in the lead.
But both candidates said they would respect the official outcome of the vote.
Three hours after voting concluded, the Electoral Institute of Mexico said the race was too close to call, and that it would not announce a winner before Wednesday.
The cliffhanger was closely followed in Washington, which evidently hoped to see a reversal of the trend that brought several leftist leaders to power in Latin America over the past few years.
The parties also waged a close race in the congressional election, and none of them won an outright majority in the lower House, according to exit polls.
The campaign of Calderon, of the ruling National Action Party (PAN), capitalized on fears among Mexican business leaders, likening the leftist candidate to Venezuela's virulently anti-US President Hugo Chavez, and warning he would plunge the country into a ruinous economic crisis.
Analysts generally dismiss the comparison and say that while Lopez Obrador, of the leftist Party of the Democratic Revolution (PRD), has little interest in foreign policy, he is not hostile to the United States, by far Mexico's largest trading partner.
Mexico's next president, who will take office on December 1, will face formidable challenges in trying to fulfill campaign pledges of battling poverty, corruption, common crime and drug-fueled violence.
Exit polls showed the ruling PAN will have 35 percent of the congressional deputies, and the PRD 31 percent. Another 28 percent of the mandates went to the Party of the Institutional Revolution (PRI) that had ruled Mexico for 71 years until its historic 2000 defeat by Vicente Fox.
Roberto Madrazo, who ran for PRI was expected to take third place in the presidential election.
Projections also showed that Mexico City will stay in the hands of the PRD, which has governed the capital since 1997, while the PAN would remain in charge of the state governments of Jalisco and Guanajuato.
Morelos, the only other state to hold elections this weekend, was too close to call, according to an exit poll.