Afghan and international officials hailed Sunday's elections as a major success in the country's march toward democracy, but chief electoral officer Peter Erben said reports from about one-third of the polling stations indicated a turnout of just over 50 percent.
International observers said that Taliban violence, which has killed more than 1,000 in the past six months, may have scared voters away.
Results were not expected for more than a week.
The government and its Western backers hailed the first elections for a national assembly in over 35 years as a strong show of defiance in the face of Taliban threats and determination to bring stability after decades of war and chaos.
"Afghanistan should be satisfied with the turnout in yesterday's election," Erben said. He said it compared well with elections in other postwar countries.
Karzai praised voters — who cast ballots in schools, mosques and even desert tents — for coming out "in spite of the terrorism, in spite of the threats." In New York, U.N. Secretary-General Kofi Annan said the election showed "the clear determination of the Afghan people to pursue the peaceful and democratic development of their nation."
President Bush called the vote successful and a major step forward, commending "the tremendous progress that the Afghan people have made in recent years."
Taliban rebels had called an election boycott. Militant attacks killed at least 15 people, including a French commando, in the hours before and during voting — the latest victims of violence that killed more than 1,200 people in the past six months.
But with tens of thousands of Afghan and foreign forces providing security, there were no spectacular assaults. Election officials said no one was killed in attacks near polling stations — although three voters were wounded — and only 16 of the 6,270 stations did not open because of security threats or logistical problems.
The voting for parliament and 34 regional councils was the last formal step toward democracy under an internationally sponsored plan laid out following the ouster of the oppressive Taliban regime by U.S.-led forces in 2001. Many people looked to a big vote to marginalize Taliban rebels whose stubborn insurgency rumbles on in the south and east.
Washington and other governments have poured in billions of dollars trying to foster a civic system that encourages Afghanistan's fractious ethnic groups to work together peacefully and ensure the nation is never again a staging post for al Qaeda and other terrorist groups.
Security was tight as workers brought ballot papers from far-flung polling stations to provincial capitals, where counting was to start Tuesday. Provisional results were expected by early October.
Once final results are posted, it will likely take time to figure out who has the power in the 249-seat Wolesi Jirga, or parliament. There are fears it could be split along the same ethnic and tribal lines that fueled years of war as 1970s coups led to a decade-long Soviet occupation followed by devastating civil war and the Taliban takeover in the 1990s.