Umar's name was announced in an online statement by al-Shabab, vowing to take revenge for Godane's death, BBC reported. The announcement came minutes after the militant group confirmed Godane's death.
The decision to appoint Godane was unanimous, an al-Shabab commander said.
Somalia's authorities earlier put the country on alert for possible retaliatory attacks by al-Shabab after the US confirmed the death of Godane in air strikes south of Mogadishu Monday night.
Meanwhile, Somalian President Hassan Sheikh Mohamoud Saturday called on Islamist militants to "embrace peace" following the death of Al Shabaab leader Godane.
Al-Shabab, meaning "the youth" in Arabic, emerged as the radical youth wing of Somalia's now-defunct Union of Islamic Courts, which controlled Mogadishu in 2006, before being forced out by Ethiopian forces.
Battling the UN-backed government in Somalia, Al-Shabab is suspected of links to a string of attacks in neighbouring Kenya.