The new government also said it would introduce a new proposal to Nile Basin nations to try to overcome a stalemate on sharing the river's waters, as a seven-minister delegation headed by Egypt's prime minister arrived on Sunday in Khartoum.
"Sudan intends to be the first to recognise Juba capital of South Sudan and Egypt intends to be the second to recognise the south," Elaraby said in Khartoum.
The delegation will visit Juba on Monday.
Egypt had pressed for its ally Sudan to remain united to preserve 1929 and 1955 Nile Basin treaties which allocated the lion's share of river waters to Egypt and its southern neighbour Sudan.
But after decades of north-south civil war, southern Sudanese this year voted to become independent on July 9.
South Sudan has not declared a position on sharing Nile waters but most analysts believe it is likely to side with its east African allies.
They have signed a new treaty aimed at ensuring what they say would be a more equitable distribution of the water, worrying Egypt which is struggling to grow food for its burgeoning population.