RW Rooble: Xil culus iyo shaqo aan badneyn!
The US carried out air strikes on Monday night targeting a convoy in which he was travelling.
"Removing Godane from the battlefield is a major symbolic and operational loss to al-Shabab," the Pentagon press secretary said in a statement.
He was one of the US state department's most wanted men.
It had placed a bounty of $7m (£4.2m) on his head.
Pentagon Press Secretary Rear Adm John Kirby said the military action on 1 September had "led to his death".
The US has supported the African Union (AU) force that has driven al-Shabab out of the capital, Mogadishu, and other towns since 2011.
The al-Qaeda-linked fighters want to overthrow the UN-backed Somali government and frequently attack government targets as well as neighbouring countries that provide troops to the AU force.
A statement from the White House press secretary described al-Shabab as the "largest al-Qaeda affiliate in Africa".
Godane's death "reflects years of painstaking work by our intelligence, military and law enforcement professionals", Jay Carney's statement said.
The al-Shabab leader had publicly claimed the group's responsibility for the deadly Westgate shopping centre attack in September last year, he said.
"Under his leadership, the group has claimed responsibility for many bombings - including various types of suicide attacks - in Mogadishu and in central and northern Somalia, typically targeting officials and perceived allies of the Somali government as well as the former transitional federal government of Somalia.
"Godane has also continued to oversee plots targeting Westerners, including US persons, in East Africa."
The White House said the US would continue to use the tools at its disposal - "financial, diplomatic, intelligence and military" - to address the threat of al-Shabab and to support the Somali government's efforts to "build a secure and stable future for the Somali people".
Ra'iisul wasaaraha ayaan haysan waqti ku filan, waxaana uu sii dhammeystiri doonaa shaqooyinka qabyada ah.