In the event of a run-off, the parties of Mr Balaguer and Mr Medina are widely expected to unite, as they did in 1996 to help the Liberation Party's Leonel Fernandez narrowly defeat the Revolutionary Party's Jose Pena Gomez.
The Central Electoral Board announced that it had suspended the vote count, with 97% of the vote counted.
Mr Mejia was left with 49.9% of the vote. He needs 50% plus one vote to avoid a run off in a second round.
The incumbent Dominican Liberation Party was a distant second with 24.84%, followed closely in third place by the Social Christian Reformist Party of former President Joaquin Balaguer with 24.68%.
A spokesman for the electoral board, Wilfredo Alemany, is reported to have said that there was no technical problem with the count, and that those counting the votes were simply exhausted and needed a brief rest.
The count was due to resume on Wednesday at 1700 local time (2100 GMT).
Observers satisfied
International observers said they were pleased with the way the poll had been conducted in the nation of eight million people.
"The Dominican people want us to govern," Mr Mejia, 59, told a crowd of cheering supporters.
Danilo Medina's free-market government has expanded the economy by almost 40% in the past four years, but many of the country's poor complain that they have seen nothing of the new wealth.
Mr Mejia, an agricultural engineer, dubbed himself "The Hope of the People," offering social programmes and small community-level public works projects.
Mr Medina insisted that the current economic boom would eventually trickle down, while Mr Balaguer promised the security of firm government.
Balaguer's influence
Mr Balaguer, now a blind and frail man, has been president of his country seven times, ruling for most of the past 40 years.
In the event of a run-off, the parties of Mr Balaguer and Mr Medina are widely expected to unite, as they did in 1996 to help the Liberation Party's Leonel Fernandez
narrowly defeat the Revolutionary Party's Jose Pena Gomez.
The early results showed Mr Balaguer's continuing influence in Dominican politics - despite the frailty reflected in his need to vote from inside his car Tuesday.
Police reported two deaths that appeared to be related to the elections, but voting was otherwise peaceful.
More than 100 foreign observers, including delegations from the Organization of American States and the Atlanta-based Carter Center, monitored the election.
About 4.25 million registered voters took part in the elections - the country's 10th since the demise of dictator Rafael Trujillo in 1961.