The video of the execution in the eastern city of Derna was posted on social media websites, including YouTube.
Amnesty International said it was carried out on August 19, apparently by an extremist militant group,the Shura Council of Islamic Youth, in front of a crowd seated in the stadium.
"This unlawful killing realises the greatest fears of ordinary Libyans, who in parts of the country find themselves caught between ruthless armed groups and a failed state," said Hassiba Hadj Sahraoui, Amnesty International's Middle East and North Africa Deputy Director.
"The Libyan authorities, with the support of the international community, must urgently address the breakdown of law and order that has persisted in Derna and elsewhere following the end of Colonel Kadhafi's rule."
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In the video, the Egyptian, named as Mohamed Ahmed Mohamed, is brought into the stadium blindfolded on a truck.
He is forced to kneel and a statement is read out accusing him of killing a Libyan, Amnesty said.
Since the fall of long-time dictator Moamer Kadhafi in a NATO-backed 2011 uprising, interim authorities have failed to establish order and security in a country prone to anarchy and deadly violence.
They have been unable to restrain a large number of militias formed by ex-rebels who fought Kadhafi and who still hold sway across Libya.
Since mid-July, Libya has been rocked by fierce and deadly fighting between militias that has prompted an exodus of foreign nationals from the oil-rich North African country.