"This is not the right time to discuss investigations, this is the time to move forward to stability and national unity," Mr Karzai said.
"I call upon our nation to change this into an opportunity to strengthen our resolve and determination, to move our country forward and to participate in the new round of elections."
Mr Abdullah told the BBC he had telephoned Mr Karzai to thank him for his remarks.
"He talked about national unity and also he stressed on the need for going to the second round which is exactly what I want to do - so that was a courtesy call, a word of thanks," he said.
He added: "I know that there are challenges with it; the security situation, and the winter is coming... but I think the fact that the process is moving now forward rather than being stuck, that in itself I consider it a step forward and we have to face the challenges."
Abdullah: 'It was an achievement'
The BBC's Martin Patience, in Kabul, says there will be no guarantee that any new vote will be free of the fraud that dogged the first round.
But for now the political deadlock appears to have been broken, for a couple of weeks at least, our correspondent says.
Mr Karzai's decision was been warmly welcomed by world leaders.
US President Barack Obama described it as an important and constructive step forward.
"It is now vital that all elements of Afghan society continue to come together to advance democracy, peace and justice," he added.
US officials later said that the president had not yet determined whether to make a decision on sending more troops to Afghanistan before the runoff.
Sen Kerry said a second round of voting was a great opportunity and a turning point, praising Mr Karzai for the "genuine leadership in the decision he has made".
French President Nicolas Sarkozy said: "President Karzai's statement shows to all that he is a statesman who can decide on what is essential, in the higher interests of his country and of the unity of the Afghan people."
UK Prime Minister Gordon Brown also welcomed President Hamid Karzai's "statesmanlike" acceptance of the run-off.
Meanwhile, correspondents say it is possible that President Karzai and his challenger may reach an agreement to form a national unity government, meaning that a run off may not be required.
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