"Even if Saddam Hussein has 100 children other than Odai and Qusai, Saddam Hussein would offer them the same way," he said on the tape.
The two were killed in a gunfight with U.S. soldiers on July 22.
The speaker also called the deaths "good news, that is the hope of every fighter for God's sake, as another group of noble souls of the martyrs have ascended to their creator."
The message came as American troops appeared to be closing in on the elusive deposed Iraqi leader.
Soldiers overpowered and arrested a bodyguard who rarely left Saddam's side Tuesday and said they obtained documents and information that could help them find the one-time dictator.
In other developments:
Tuesday's broadcast, monitored in Cairo, was the second tape attributed to Saddam in just over a week.
The voice began the tape with a verse from the Quran and said those killed in the gunfight in the northern Iraqi town of Mosul would be martyrs in heaven.
Like the tape played Tuesday, the previous audio recording attributed to Saddam contained a topical reference that suggested when it was recorded.
That recording said he was speaking on July 14 and referred to the new Governing Council of Iraq.
U.S. intelligence officials said the earlier recording was probably authentic and was further evidence that Saddam survived the war.
However, the poor quality of the recording prevented technical analysts from being absolutely certain of its authenticity, the official said, speaking on the condition of anonymity.
The bodyguard nabbed Tuesday, Adnan Abdullah Abid al-Musslit, was caught in a series of pre-dawn raids in the heart of the deposed dictator's hometown that nabbed a total of 12 people, including a leader of the Saddam Fedayeen militia.
At around 4 a.m., soldiers fired three shotgun blasts into the locks of the house where bodyguard al-Musslit was living with his family.
The stocky bodyguard struggled to break free as soldiers arrested him, and they had to wrestle him to the ground and drag him down the stairs, Lt. Col. Steve Russell said.
It was the latest operation in a stepped-up hunt for Saddam, following raids over the weekend and last week's capture of several other bodyguards for the deposed leader.
The 22nd Infantry said it came within 24 hours of catching Saddam's new security chief — and possibly the dictator himself — at a farm in eastern Tikrit on Sunday.
It also assisted Task Force 20, the special operations force hunting Saddam, in a Thursday raid that netted up to 10 of the dictator's suspected bodyguards in al-Uja, the Tikrit suburb where Saddam was born.
The search for Saddam turned bloody Sunday when a U.S. raid in Baghdad's upscale Mansour district left at least five Iraqis dead. Britain's Guardian reports the incident has sparked anger against the United States.