In a letter to President Hamid Karzai, Mr Obama expressed his "deep regret" and said the incident earlier this week was a genuine mistake.
Demonstrations against the desecration have continued for a third day across northern and eastern Afghanistan.
Two US soldiers and two Afghans were killed in an attack on a military base. Elsewhere there were four other deaths.
On Wednesday, another seven people were killed and dozens injured in protests.
Mr Obama's letter, delivered by the US ambassador to Afghanistan, assured the Afghan president that US authorities would question all those responsible.
"I convey my deep sympathies and ask you and the people to accept my deep apologies," the letter said."The error was inadvertent; I assure you that we will take the appropriate steps to avoid any recurrence, to include holding accountable those responsible," it continued.
President Karzai told members of the Afghan parliament that a US officer was responsible for the burning.
But he said it was done out of "ignorance".
Thursday, February 23, 2012
Barack Obama apology to Afghanistan over Koran burning
Which again leaves me to apologize for Obama. I missed it, when did he apologize to the Catholics for forcing Obamacare on them? When did he apologize to the Jews for calling them wrong for trying to protect their people?