The London-based human rights organisation also urged the military authorities to ensure that new emergency laws do not undermine civil liberties.
"Bashir is wanted for some of the most odious human rights violations of our generation, and we need to finally see him held accountable," Amnesty secretary general Kumi Naidoo said in a statement.
"The Sudanese authorities should now turn Bashir... over to the International Criminal Court so that the victims of these unspeakable crimes can see that justice is done."
The 75-year-old is wanted by the ICC in The Hague on charges of genocide and war crimes.
Amnesty said it was "alarmed" by the new emergency measures that have been installed under the incoming transitional military council.
"Sudan's military authorities should ensure that emergency laws are not used to undermine people's rights," said Naidoo.
"The transitional authorities must take all necessary measures to facilitate a peaceful transfer of power in Sudan. That means respecting the rights to freedom of expression and assembly."
Naidoo said the world should recognise the courage Sudanese people have shown in demanding their civil liberties.
He added: "Today's events should also serve as a wake-up call to leaders around the world who think they can get away with denying people their basic rights."