Hussein Yahya Jangol was named to head the central bank, a presidential decree said.
Jangol, who replaces Mohamed Khair al-Zubair, has worked with the central bank since 1982.
The change comes as the central bank grapples with an acute shortage of foreign currency, a key factor in triggering nationwide protests against the veteran leader's rule stretching back three decades.
Bashir has launched sweeping changes in face of the protests, including imposing a year-long state of emergency across the country and dissolving the federal and provincial governments.
He has also handed over his own powers as the chief of the ruling National Congress Party to his deputy in the party, Ahmed Harun.
Deadly protests erupted in December after a government decision to triple the price of bread but they swiftly mushroomed into nationwide demonstrations against Bashir's rule.
Protesters accuse Bashir of mismanaging the economy and have called on the 75-year-old president to step down.
Officials say 31 people have died in protest-related violence, while Human Rights Watch has put the death toll at at least 51.
Earlier on Wednesday, parliament began discussing the state of emergency Bashir imposed on February 22.
Bashir has remained defiant in the face of protests and declared the state of emergency after an initial crackdown failed to halt the demonstrations.
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