Moncef Marzouki has refused to admit defeat in Tunisia's first free presidential election after exit polls suggested Beji Caid Essebsi had won.
The caretaker president, a former exile, said his rival's declaration of victory was "undemocratic".
Mr Essebsi, 88, has urged all Tunisians to "work together" for stability.
Critics say his success marks the return of a discredited establishment, as he served under deposed President Zine el-Abedine Ben Ali.
Mr Ben Ali was ousted in 2011 after the Arab Spring revolution that triggered uprisings across the region.
Mr Essebsi was also in the cabinet of Tunisia's first post-independence leader, Habib Bourguiba.
Official results are not expected until Monday evening. One exit poll gave Mr Essebsi 55.5%, with several others showing similar figures.
The election marks the last stage of Tunisia's move to democracy, after the revolution.
Analysis: Naveena Kottoor, BBC News, Tunis
Moncef Marzouki's response to Beji Caid Essebsi declaration of victory perhaps sounded rather indignant. But Tunisia's incumbent president clarified on Sunday he would accept the official results once they had been announced by the election authorities.
Observers who were deployed across the country in their thousands have not raised serious concerns so far. Should that change, there are legal systems in place for either candidate if they wish to challenge the result.
And while this election is an historic achievement, it is important to remember that the new democratically elected president will be less powerful than his predecessors. The experience of dictatorship is still fresh in the minds of many Tunisians and the founding fathers of the new Tunisia made sure the new president would not have similar powers.
It is the first time Tunisians have been able vote freely for their president since independence from France in 1956.
Mr Marzouki, a 67-year-old human rights activist forced into exile by the Ben Ali government, said the election was too early to call.
"The announcement of victory is undemocratic and we should wait if we want to be a state that respects the rule of law," he told supporters.
"What I want to tell you is that we are victorious, we are victorious, we are victorious. Tunisia has won and you have won. You have won for Tunisia, for democracy and for human rights."
'Martyrs of Tunisia'
Mr Marzouki has been interim president since 2011 and is more popular in the conservative, poorer south.
He was thought likely to attract support from the moderate Islamist Ennahda party, which has played a key role in Tunisian politics since the Arab Spring but did not field a candidate.
Mr Essebsi appeared on television after polls closed on Sunday, saying: "I dedicate my victory to the martyrs of Tunisia."
"I thank Marzouki, and now we should work together without excluding anyone," he added.
Beji Caid Essebsi
- 88-year-old lawyer and politician
- Studied law in Paris
- Interior minister under Habib Bourguiba, Tunisia's first president after independence
- Speaker of parliament under ousted President Zine el-Abedine Ben Ali
- Interim prime minister in 2011 after the uprising
- Founder of secular-leaning Nidaa Tounes party in 2014
- Supported by Tunisian General Labour Union and some business groups
Supporters danced and let off fireworks outside the headquarters of Mr Essebsi's secular-leaning Nidaa Tounes party.
Mr Essebsi led in the first round of voting last month with 39% of the vote. Mr Marzouki polled 33%.
He is popular in the wealthy, coastal regions, and based his appeal to voters on stability and experience.
Whoever wins faces restricted powers under a constitution passed earlier this year.
The president will be commander-in-chief of the armed forces but can appoint or sack senior officers only in consultation with the prime minister.
The president will also set foreign policy in consultation with the prime minister, represent the state and ratify treaties.
Tunisia boosted security for the elections and closed border posts with Libya, which has been plagued by unrest.
A group of at least three attackers targeted a polling station near the city of Kairouan on Sunday morning. Security forces say they killed one attacker and arrested three.