He writes that Bin Laden got a secret communication and fled his Afghan camp two hours before cruise missiles struck it.
He does not identify the source of the tip, which the U.S. suspects was Pakistani intelligence.
Omar's book, "Growing Up Bin Laden," written with his mother, Najwa - the Al Qaeda leader's first wife - describes the ultimate dysfunctional family.
The Bin Ladens lived austerely as their father staked his horrific claim as the world's most wanted man. His son eventually concluded Bin Laden hated his enemies more than he loved his family.
Omar, 28, describes weeping as a teenager when told that Al Qaeda needed his pets to conduct chemical warfare tests.
"After I learned the truth about the puppies, I turned even further away from my father," whose jihad led only to death, Omar writes in the book set for release by St. Martin's Press later this year.
It has been widely reported that Bin Laden's goons tested nerve agents at the Derunta camp in Afghanistan. In 2002, CNN obtained and showed video of dogs - fully grown - being gassed by visible toxic fumes.
Bin Laden's fourth son admits he knew in advance of plots against targets like the 1998 bombings of U.S. embassies in East Africa, where 224 perished.
He called the 9/11 attacks "horrific." They occurred after he was told by his best friend - Al Qaeda operative Abu al-Haadi - that a "new mission" would be much bigger than the embassy bombings. Omar mourned al-Haadi's death in the resulting U.S. invasion of Afghanistan.