As the ballots were being collated in Kabul, there were reports of fresh violence in Jalalabad, the capital of Nangarhar province.
At least one suspected Taliban fighter was reportedly killed during a battle with security forces around a police station in the city.
Abdullah, Karzai's former foreign minister, said that he was on track to win the vote, which went ahead despite numerous Taliban attacks.
"So far it is very good," he told Al Jazeera.
"Of course, we will let the election commission announce the results when they have all the results from the south ... but in the areas where the results are announced I am in the lead, no doubt."
Waheed Oner, a spokesman for Karzai, said that his camp's belief that it was heading to victory came from media reports of results already announced by the election commission.
"We are in the lead, but being in the lead is not equal to having won the elections," he told Al Jazeera.
"We expect to win the elections but obviously we will have to wait for the results to be announced."
Fazl Sangcharaki, Abdullah's spokesman, said the north of the country had voted solidly for Abdullah, except in Jowzjan province, home of Abdul Rashid Dostum, an Uzbek leader who returned to the country days before the vote to campaign for Karzai.
However, analysts have said that the eventual result is likely to swing on turnout in the south of Afghanistan, where Karzai, an ethnic Pashtun, garners most of his support.
|At least 26 people were killed in 135 violent incidents on polling day [EPA]
Al Jazeera's Zeina Khodr, reporting from Kandahar, said that although officials were pleased with the turnout many people had not voted because of their concerns about possible attacks.
"On election day, Kandahar was rocked by explosions every half hour from rocket fire landing in the city," she said.
"But the real question is what took place in remote districts where the Taliban is stronger? We went to one today ... and we talked to people there and they did not vote because of security."
The Taliban had warned Afghans not to vote and threatened to cut off the fingers of those who voted and launch attacks on polling stations.
Nine civilians, nine policemen and eight soldiers were killed in 135 incidents across the country, officials said. Twenty-eight other soldiers were wounded and security forces reportedly thwarted five suicide attacks.
The International Republican Institute, a US-based group that monitors elections, said Thursday's vote was marked by "serious problems", but had so far been credible.
Afghanistan's election authority said it was launching an investigation into complaints of irregularities.
"We have received complaints but I don't think that is going to be material to the results"
EU election monitor
Abdullah released a statement detailing about 40 incidents of alleged irregularities, most claiming officials had pressured voters into choosing Karzai.
There were also claims polling stations did not have enough ballot papers and that the indelible ink used to mark the fingers of voters as a safeguard against multiple voting could be washed off.
"I think there were some problems as to how the ink was applied initially, but the IEC moved on that fairly quickly I am told, advising their polling stations workers to shake the bottle," Grant Kippen, the chairman of the European Union's electoral complaints commission and an election monitor, said.
"We have received complaints but I don't think that is going to be material to the results," he told Al Jazeera.
Barack Obama, the US president, on Friday hailed the elections and pledged to work with the new government.
"This was an important step forward in the Afghan people's efforts to take control of their future, even as violent extremists
are trying to stand in their way," he said.