"As the Emir of al Qaeda jihadist group, I present our allegiance to you," say the man's voice, which sounds like al-Zawahiri's.
It is believed that the statement was recorded on May 27 and it has been released through social media channels.
Akhundzada was made the chief of the Afghan militant group last month after his predecessor Mullah Akhtar Mohammad Mansour was killed in a drone strike in Pakistan.
The man purporting to be al-Qaeda said the jihadist group was "sad" over the death of Mansour. He showered praises on Mansour for fighting against what he called "Russian and Christian crusaders" and for "dying a martyr after a long history of jihad."