"I'm happy to be in Khartoum after 22 years (away)," Kiir told reporters after he flew in from southern Sudan on Wednesday with security tight in the capital.
"Despite the fact that we have lost our hero, the man who brought peace, Dr Garang, we will continue with the same vision, with the same objective, and we will implement the Comprehensive Peace Agreement."
He was referring to the peace deal signed with the Khartoum government in January that ended 21 years of north-south war in Sudan that cost two million lives.
Kiir urged Sudanese not to resort to violence after deadly rioting shook Khartoum and several towns in south Sudan following Garang's death on 30 July in a helicopter crash.
"I'm appealing to all the people of Sudan, the people from southern Sudan in particular, to remain calm and not to repeat all what has happened a few days ago," he said.
Kiir was greeted by Sudan's second vice president, Osman Ali Taha, as he stepped out of the plane after flying in from Rumbek in south Sudan.
About 200 Sudanese dignitaries, including officials from Kiir's Sudan People's Liberation Movement/Army (SPLM/A) and the ruling National Congress Party, shook hands with Kiir inside the airport.
President Omar al-Bashir (L) and
Salva Kiir (R) at Garang's funeral
The authorities stepped up security in the capital and its suburbs ahead of the visit, deploying additional police and soldiers on the streets.
Army troops were seen lining the main avenue from downtown Khartoum to the airport, two miles away.
There were no posters of Kiir on the streets or public buildings as was the case with the high-profile return to Khartoum of his predecessor on 8 July, when more than a million people showed up at the city's main square to welcome Garang.
Kiir is to be sworn as first vice president at a brief ceremony in Khartoum on Thursday, replacing Garang, who died three weeks into the job.
One of the first challenges for Kiir, the former southern rebel military chief, will be to form a national unity government with former arch-foe President Omar al-Bashir - a process interrupted by Garang's death.
The reconciliation cabinet and an autonomous government in south Sudan were due to be installed by 9 August, in line with the January peace deal.
The United States, which invested much political capital into securing the 9 January pact, has expressed confidence in the SPLM/A and Kiir's ability to uphold the deal.