A member of the group calling itself Usd Allah (God's Lions) said Qutb was freed because he was "loyal to Islam and had good manners".
Aljazeera reported that the group had refused to accept large sums of money in return for freeing the captive.
Qutb was seized in Baghdad after Friday prayers in response to comments by Egyptian Prime Minister Ahmad Nazif that Egypt was ready to offer its security experience to the interim Iraqi government, a videotape aired on Aljazeera channel said.
The Egyptian government later clarified it was "absolutely not considering" deploying troops in Iraq.
Egyptian Foreign Minister Ahmed Abu al-Ghait on Monday expressed gratitude to the Iraqi people for helping to secure Qutb's release.
"I thank the Iraqi people who contributed towards the release of the Egyptian diplomat," he said.
Abu al-Ghait said the diplomat, currently at the Egyptian embassy in Baghdad, was in good health.
He said he had also congratulated Qutb's family on his safe release which Abu al-Gheit said followed "intense contacts" by the Egyptian side with different Iraqi officials.
He added the freeing of Qutb was likely connected to his work on promoting "brotherly relations between the Iraqi and Egyptian people".