There were reports of civilian deaths overnight in clashes between southern and northern Sudanese in the suburbs. Preparations were underway to bury Garang, who died on Saturday in a helicopter crash near the Ugandan border.
Sudanese television said some people had burnt business premises in Khartoum’s northeastern Haji Yusuf suburb, but police had been deployed there to contain the situation.
The death toll from the violence - which erupted on Monday after news of Garang’s death - had reached 46 by Tuesday, the Deputy Commissioner-General of Police, Lt Gen Husayn Uthman, said in a statement.
Another 360 people had been injured, 65 vehicles damaged and 45 business premises destroyed.
The southern Sudan People's Liberation Movement/Army (SPLM/A), which Garang had led through two decades of war against the Sudanese government, called for calm and vowed to continue the Sudanese peace process.
Speaking to reporters on Tuesday in the Kenyan capital, Nairobi, SPLM/A spokesman Samson Kwaje said Garang would be given a state funeral in the southern town of Juba on 6 August.
Thousands of Sudanese and foreign guests had on Tuesday arrived to pay their condolences at New Site in Kapoeta, southern Sudan, where Garang's remains were flown from the crash site.
Kwaje said the SPLM/A council, at a meeting at New Site, had resolved to move the headquarters of the movement from Rumbek to Juba.
The council also nominated the new SPLM/A chairman, Salva Kiir Mayardit, for appointment as First Vice President of Sudan and President of southern Sudan, in accordance with the constitution.
The new SPLM/A chairman was due to meet two senior United States envoys, Constance Newman and Roger Winter, at New Site on Wednesday afternoon.
Sudan's ruling National Congress Party welcomed Salva Kiir's appointment, and expressed its willingness to cooperate with the new SPLM/A leader.
In New York, the UN Security Council urged the Sudanese people to be calm, a UN statement said on Tuesday.
Ambassador Kenzo Oshima of Japan, which holds the Council's rotating presidency for August, said the Council felt Garang's death should not "deter the Sudanese peoples' struggle for justice and dignity".
It also encouraged "all the people of Sudan to refrain from violence and to maintain peace in the midst of mourning".
Oshima said the Council trusted that the Sudanese people would remain united and continue to work to implement the Comprehensive peace Agreement (CPA).
The CPA, which was signed by Garang and Sudanese President Umar al-Bashir on 9 January in Nairobi, ended two decades of conflict in southern Sudan in which millions were killed or displaced.
It paved the way for Garang's appointment as First Vice President of Sudan and President of southern Sudan - positions to which he had only been appointed on 9 July.
Garang died en-route to southern Sudan from Uganda, following a meeting with President Yoweri Museveni. Sixteen other people who were travelling with him in the Ugandan military helicopter also died.