The claim, made Sunday, came a day after the release of a video announcing the murder of security contractor Haruna Yukawa by the jihadist group, which has beheaded five Western hostages since August last year.
It sought to raise the pressure on Japan by saying the fate of the other captive, freelance journalist Kenji Goto, depended on the release of an Iraqi would-be female bomber who is on death row in Jordan.
"The Islamic State has carried out its threat... it has executed Japanese hostage Haruna Yukawa after the expiry of the deadline given," the jihadist group said on its Al-Bayan radio.
"The second hostage is calling on his relatives to put pressure on the (Japanese) government for the release of our sister Sajida al-Rishawi, held in the jails of the oppressors in Jordan, in exchange for his release."
Rishawi was sentenced to death by a Jordanian court in September 2006 in connection with triple hotel bomb attacks in Amman the previous year, which killed 60 people.
Her name emerged Saturday in an IS video that showed Goto holding what appears to be a photograph of Yukawa's slain body.
The audio recording was of a man claiming to be Goto who says Yukawa was killed because Japan had failed to pay a $200 million ransom within a 72-hour deadline announced Tuesday.
The video unleashed a tide of global revulsion while Yukawa's father voiced horror and shock.
"I thought 'Ah, this finally happened' and was filled with regret," said Shoichi Yukawa.
"I went totally blank, I was only sorry... I had no words," he said. "In my mind I wish very much that this wasn't true."
Japanese Prime Minister Shinzo Abe branded the execution "outrageous and unforgivable" and called for Goto's immediate release.
"I condemn it strongly and resolutely," said the Japanese leader.
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US President Barack Obama condemned the "brutal murder" and offered Abe condolences while expressing his solidarity with the people of Japan.
British Prime Minister David Cameron decried the Islamic State group's "murderous barbarity", and French President Francois Hollande labelled it a "barbaric assassination".
The United Nations Security Council unanimously condemned the group's "brutality" and its "heinous and cowardly act".
Germany and the EU also condemned the killing by IS, an extremist Sunni Muslim group which activists say carries out near-daily executions, often beheadings, in areas under its control.
The group, which originated in Iraq and appeared in Syria's civil war in spring 2013, holds swathes of Syria and Iraq where it has declared an Islamic "caliphate".
Since then it has committed atrocities including the beheadings of two US reporters, an American aid worker and two British aid workers.
The Amman bombings -- including one that hit a hotel where a wedding party was in full swing -- were claimed by Jordanian-born Abu Musab al-Zarqawi, the Al-Qaeda leader in Iraq who was killed in a US air raid there in June 2006.
His group was a precursor of IS, and Rishawi's brother, Samir Atruss al-Rishawi, who was also killed in Iraq, was one of Zarqawi's top lieutenants.
Sunday's radio statement made no mention of the $200 million ransom that the group had initially demanded, but only referred to the release of Sajida al-Rishawi.
The 44-year-old woman has been sentenced to death in Jordan, which lifted an eight-year moratorium on capital punishment in December by hanging 11 men convicted of murder.
The ransom demand came as Abe pledged, during a trip to the Middle East, a multi-million dollar aid package to countries affected by the militant group's bloody expansion in Iraq and Syria.
But Abe has been defiant.
"We will never give in to terrorism, and we will actively contribute to the peace and stability of the world together with the international community. We are not wavering at all on this policy," he said Sunday.