US and Nato military officers said there were no reports of attacks by Taliban fighters, who have vowed to disrupt what they call a US-orchestrated sham.
Polls opened at 7am (0230 GMT) on Saturday and are to close at 4pm.
The focus is on how many people turn out in defiance of the Taliban and whether US-backed favourite President Hamid Karzai gets the 51% vote he needs to avoid a November runoff.
In Kabul there were just a handful of voters outside main polling stations in the morning.
By midday, the main election-related problem stemmed from workers using the wrong pen to mark people’s fingers after they voted. The ink was not indelible and could be washed off, leading to an uproar among several candidates concerned about possible multiple voting.
Concerns were raised briefly on Saturday when a Western official reported a bombing at a polling station in the north, but Afghan officials and international peacekeepers said no such blast had occurred. They said a truck fire had broken out but caused no injuries.
A flurry of rockets landed in several cities around the country on Thursday and Friday, including one that hit a parking lot near the US embassy and another that injured a young girl and an old man in the eastern city of Jalalabad.
Hamid Karzai needs 51% of vote
Karzai’s team is reportedly hoping that at least 60% of the almost 12 million eligible voters will turn out.
That would go a long way to prove that democracy has made a strong start in a nation invaded by US-led forces three years ago and still controlled in large part by regional commanders.
“Whichever government is made by the Americans for us, it will be unacceptable to us at any cost,” Mulla Ubaid Allah, the former Taliban defence minister, said in a statement on Friday.
“The Americans have insulted our autonomy, independence and dignity (and) unjustly invaded our frontiers. We will retaliate to this insulting attitude and treatment with full force.”
A policeman was slightly hurt when grenades were thrown at a polling station for Afghan refugees near the Pakistani city of Peshawar before voting began on Saturday, police said.
The two grenades exploded at a school on the outskirts of Peshawar shortly after midnight (1900 GMT Friday), nearly seven hours before the polls opened.
Most observers accept that Karzai will win, probably immediately or otherwise after a runoff, and believe the main challenge is in conducting a smooth election in the landlocked, impoverished nation of 28 million people.
Refugee Muqadasa Siddiqi, 19,
About 18,000 US-led troops, hunting al-Qaida and Taliban leaders, are helping a 42,000-strong Afghan police and military force and 8000 Nato-led peacekeepers to provide security.
They were on full alert as the nation prepared to vote.
The first person to vote in the historic election was a 19-year-old woman – but she was in Pakistan.
As in Afghanistan, polls for hundreds of thousands of Afghan refugees living in Pakistan also opened at 7am. But time in Pakistan is half an hour ahead of that in their homeland, so the first vote was cast at 0200 GMT.
“I cannot explain my feelings, just how happy I am,” said Muqadasa Siddiqi, a science student whose family escaped Kabul in 1992 during civil war.
“I would never have thought I would be able to vote in this election.”
In Kandahar police struggled to keep back a crowd of hundreds of men trying to vote at a polling site on the grounds of the governor’s residence.
Election workers reprimanded several Karzai campaign officials for coming to lobby voters at the site, a violation of electoral law.
Donkeys will be used to transport
Voters who came out early across the country said they hoped the election would put an end to their nation’s long suffering.
“We are very happy,” said Abd al-Rahim, a 75-year-old voting in the northern city of Sheberghan, a stronghold of Uzbek regional commander Abd al-Rashid Dustum, one of Karzai’s challengers. “We want the election to end our wars.”
Counting will begin immediately after polls close in Afghanistan and first trends will be clear by Monday.
But a full count will not be available until late October – donkeys are being used to bring down ballots from some of the mountains in the Hindu Kush – and it is likely Karzai will not know until then whether he needs to contest a second round.
Eighteen candidates are in the fray although two announced earlier this week that they were withdrawing in favour of Karzai.
While Karzai is easily the best-known candidate, the patchwork of ethnic groups that make up Afghanistan could work against him if people vote on community lines. He is a Pashtun, the group to which the country’s rulers have traditionally belonged and makes up about 40% of the population.
Minorities include the Tajik, Hazara, Uzbek and Baluch communities.