More than 50 percent of registered voters - about seven million people - were estimated to have turned out on Saturday for local and presidential elections, which will usher the country's first democratic transfer of power.
"It is a proud day for this proud nation. We have proven that we are people of the ballot, not of the bullets," said Ashraf Ghani Ahmadzai, a presidential candidate. "This is a day of celebration."
The outgoing Afghan president, Hamid Karzai, said: "Today's election and massive participation of the people have taken Afghanistan a few steps forward to peace, stability and development."
Barack Obama, the US president, congratulated the Afghan people and said the event was a milestone as the Afghan people take full responsibility for their country.
Preliminary results for the presidential poll are not due until April 24 and a run-off election is scheduled for May if there is no outright winner.
There is no clear favourite but front-runners include Zalmai Rassoul, a former foreign minister; Abdullah Abdullah, the runner-up to Karzai in the 2009 election; and Ghani, a former World Bank academic.
Whoever succeeds Karzai must lead the fight against the Taliban without the help of US-led troops, who are due to pull out by the end of 2014, and also strengthen an economy that currently relies on aid money.
The Taliban had urged their fighters to target polling staff, voters and security forces, but there were no major attacks reported during the day.
Thousands lined up outside polling centres in Kabul, which has been hit by a series of deadly attacks during the election campaign.
"I'm not afraid of Taliban threats, we will die one day anyway. I want my vote to be a slap in the face of the Taliban," housewife Laila Neyazi, 48, told AFP.
One blast in Logar province, south of Kabul, killed one person and wounded two, according to Mohammad Agha district chief Abdul Hameed Hamid.
Omar Daudzai, the interior minister, said said four civilians, nine police and seven soldiers had been killed in violence on election day, and added that many attacks had been foiled, without giving further details.
Attacks or fear of violence had forced more than 200 of a total 6,423 voting centres to remain closed.