The BBC's Andrew North in Kabul says a key reason for holding a run-off vote was to try to restore some legitimacy to the election process after the widespread fraud in the first round.
But with the result now a foregone conclusion, our correspondent says the UN, the US and the UK want to see the election brought to an end and President Karzai declared the winner.
'National unity'
Mr Ban had welcomed the run-off when it was announced in October.
He told the BBC last month that the widespread fraud in the first round was "painful" and he hoped it would not be repeated on 7 November.
The presidential election run-off is now a one-horse race
Mr Ban said the UN would take all "necessary measures, administrative and security" to ensure a properly conducted vote.
Dr Abdullah told the BBC on Sunday he had made the decision to withdraw "in the best interests of the country". Earlier, he had told supporters his demands for ensuring a fraud-free election had not been met.
President Karzai says the Afghan election commission should decide what happens now. He has rejected Mr Abdullah's demand that election officials who presided over the first round be dismissed.
US Secretary of State Hillary Clinton said she respected Dr Abdullah's choice and urged him to still work for peace.
The US would continue to support both the people of Afghanistan and their next president, Mrs Clinton said.
UK Prime Minister Gordon Brown said Dr Abdullah had pulled out of the election "in the interests of national unity" and that President Karzai now wanted to issue a "unity manifesto".
Fraud
An investigation by the UN-backed Electoral Complaints Commission (ECC) after the first round of polling saw Mr Karzai's share of the vote drop to 49.67% - below the crucial 50% plus one vote threshold needed to avoid a second round.
Dr Abdullah was adjudged to have won about 31% of valid votes cast.
The Tajik-Pashtun former eye surgeon served as foreign minister in a short-lived government headed by the Northern Alliance.
Dr Abdullah continued as "foreign minister in exile" throughout the years of Taliban rule, which ended in 2001.
He resumed the role in the government of President Karzai after the fall of the Taliban, before leaving it five years later.
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