CBS News senior correspondent John Miller, a former assistant FBI director, cited sources as saying that the 19-year-old Tsarnaev had written the message with a pen on the inside wall of the boat cabin as he hid from police, bleeding from gunshot wounds sustained in a shootout with police that killed his 26-year-old brother Tamerlan.
The text says the bombings were "in retribution for the US crimes against Muslims in places like Iraq and Afghanistan, that the victims of the Boston bombing were collateral damage, the same way innocent victims have been collateral damage in US wars around the world, summing up that when you attack one Muslim, you attack all Muslims", Miller said.
Miller described the message as a "claim of responsibility" and said the writings are "certainly statements that are admissible" in court.
He said the statements presented prosecutors with a clear picture of the Tsarnaev brothers' motive, and that the note was consistent with what Dzhokhar Tsarnaev had told investigators while in custody.
The Tsarnaev brothers, Russians with Chechen roots who were living in Massachusetts, allegedly planted two pressure-cooker bombs laden with shrapnel and explosives near the finish line of the April 15 Boston Marathon, killing three people and wounding more than 260 others.
US authorities have charged Dzhokhar Tsarnaev with using a weapon of mass destruction against people and property. The White House has rejected demands by some Republican lawmakers that he be tried before a military tribunal as an "enemy combatant".
If convicted, Tsarnaev could face the death penalty, according to a statement by the US Justice Department.