The affair - dubbed "Guptagate" after the influential India-born Gupta family - has transfixed South Africa since the private flight landed at Pretoria's Waterkloof Air Force base last month with nearly 200 guests for a lavish family wedding.
It has also shone an uncomfortable spotlight on Zuma's relationship with Gupta brothers Atul, Ajay and Rajesh, whose South African business empire stretches from mining and resources to aviation and technology.
Justice Minister Jeff Radebe said the Gupta family had resorted to using diplomatic channels after their initial requests for a special reception at Johannesburg's main airport had been repeatedly denied.
"An individual in the High Commission communicated directly with individuals at the air force command post. The collusion of officials resulted in the irregular approval of the flight clearance," he told a news conference attended by a number of government ministers, summarising the findings of an investigation into the matter.
"The landing of the chartered commercial aircraft, flight JAI 9900, was a direct result of manipulation of processes."
Other people implicated in the affair at Waterkloof, one of South Africa's most important military sites, include Bruce Koloane, a top foreign ministry bureaucrat, and a Waterkloof movement control officer who helped secure the landing rights.
Neither official was available for comment.
Details of the report will be released next week.
Officials at the Indian High Commission could not immediately be reached for comment. A Foreign Ministry spokesman in New Delhi said this was an internal South African issue and declined to comment further.
So far India has been saying that no protocols had been breached and that the plane was carrying politicians from the Samajwadi Party, a regional ally of the Delhi government led by prominent national politician Mulayam Singh.
South African Foreign Minister Maite Nkoana-Mashabane said relations with India remain "solid and sound". The findings of the report will be shared with the Indian authorities, she said.
Radebe said several criminal cases have already been opened, and he expects others to follow as the investigation unfolds.
He said the preliminary findings already showed the name of the president and two other government ministers had been used to exert influence on civil servants.
Even so, the episode has so far failed to suck in Zuma, who missed the wedding due to a state visit to the Republic of Congo, but it is seen as another example of the influence-peddling and cronyism that the president's critics say has become endemic since he came to power in 2009.
Two of Zuma's children have served as directors of a number of Gupta companies, according to South Africa's companies database, and the family is a major financial backer of the ruling African National Congress (ANC).
In addition to several suspended base commanders, some policemen had been arrested for moonlighting as private security guards in the luxury car convoy that took the wedding guests from the air base to Sun City.
The police have also opened a criminal case against a Pretoria car hire firm for the illegal use of flashing 'blue lights' in the suite of BMWs in the convoy.