Security has been tightened across the country, with tens of thousands of troops and police deployed to guard polling stations and prevent attacks.
Early on Saturday, an explosion occurred near a polling station in the southern city of Kandahar, wounding at least 15 people.
The polls opened at 7am local time (2:30 GMT) and are scheduled to close at 5pm (12:3 GMT).
The two men have shared power over the past five years in a so-called unity government formed by the United States in the wake of allegations of widespread fraud and corruption in the 2014 polls.
The Taliban, which has been waging a war demanding the withdrawal of foreign troops from the country following its 2001 overthrow by US-led forces, has denounced the election as a “sham”.
On Thursday, the group urged Afghans to boycott the vote and threatened to attack security forces, block roads and target polling stations across the country.
“We ask fellow countrymen to refrain from venturing out of their homes on this day so that, may [God] forbid, no one is harmed,” it said in a statement.
The Taliban began peace negotiations with the US in October last year but earlier this month, following the conclusion of the ninth round of talks, US President Donald Trump dashed increased hopes of an agreement by abruptly declaring the discussions as “dead”.
‘A leader to end the war’
More than 72,000 security personnel have been deployed to 49,402 polling booths nationwide, while 410 polling centres will remain closed on Saturday over security concerns.
The threat of attacks has been on Afghan voters’ minds, but some said they are still determined to go out and vote.
“These elections are important to us because we want a leader who will negotiate peace with the Taliban and end the years-long war in the country,” Ismatullah Safi, a taxi driver in the capital, Kabul, told Al Jazeera.
In the lead-up to Saturday’s vote, the fragile security situation and the ailing economy appeared to dominate voters’ concerns.
Meanwhile, Abdullah, 59, has accused Ghani, 70, of abuse of power and using government resources in his election campaign – allegations Ghani denies.
According to the Independent Election Commission, 9.6 million Afghans, including some 3.3 million women, have registered to vote.
A presidential candidate must secure 50 percent of the vote to win outright. If no candidate crosses the threshold, a runoff will be held between the top two contenders – most likely on November 23.
Preliminary results are not expected before October 17, and the final results will be announced on November 7.