Ramazan Bashardost, a popular Afghan MP who camps in a tent near parliament and campaigned against corruption, got 53,740 votes, while Ashraf Ghani, a former World Bank economist, got 15,143.
James Bays, Al Jazeera's correspondent in Kabul, said the results that had been released were "non-contested".
"I must stress these are preliminary figures," he said. "This all could change. We are going to get many, many more results.
"I spoke to one of the key figures in the election commission who said they wanted the Afghan people to have some confidence in this process and that is why they released this batch of figures. From now on, every day, fresh figures will be released."
As the partial results were released, the number of complaints of election fraud continued to rise.
The UN-backed Electoral Complaints Commission said on Tuesday that it had received 790 complaints since election day.
Fifty-four of those cases were given "high priority" and were considered serious enough to potentially affect the result, the group said.
The complaints had more than tripled since Sunday when 225 objections had been registered.
Abdullah and other challengers have accused Karzai and the authorities of widespread fraud.
"We will not allow big fraud to decide the outcome. There is no doubt state-crafted widespread fraud is under way," Abdullah told reporters before the partial results were released.
Abdullah said he had no plans to make a deal with Karzai to drop his bid for the presidency.
"I will not make deals with anybody ... I will defend your votes," he said. Asked if the election dispute could lead to violence, he added: "I hope not."
Karzai needs a simple majority of more than 50 per cent to avoid a second round run-off. Otherwise, a run-off is planned at the beginning of October.