The complaint, alleging graft against the brothers who made their fortune in South Africa after emigrating from India in the early 1990s, also names one of Zuma's sons, Duduzane, a former Gupta business partner, and Mineral Resources Minister Mosebenzi Zwane.
"Economic Freedom Fighters (EFF) opened a criminal case against the Gupta-led criminal syndicate at the Rosebank Police Station on the 16th of October," a statement by the radical leftist party said.
"The case relates to instances of corruption, theft, fraud, money laundering, racketeering and various offences under the Income Tax Act, Financial Intelligence Centre Act and the Currency Laws of the Republic of South Africa," it said.
The party had also submitted a sworn statement from Finance Minister Pravin Gordhan "as a basis of proof that there is a prima facie case against the Gupta criminal syndicate", the statement said.
Zuma has been under increasing pressure over persistent allegations the powerful family held undue political sway over him to the extent it could even nominate a cabinet minister.
But he has denied any abuse of influence emanating from his links to the family -- who preside over a business empire with interests in mining, transportation, technology and media -- to whom he admits he is close.
The complaint, which names a total of 13 people and businesses, was filed just days after Gordhan filed court papers containing details of exchanges he had with South Africa's Financial Intelligence Centre (FIC) regarding "suspicious" transactions made by firms linked to the Guptas over the past four years.
In his deposition, which was seen by AFP on Saturday, Gordhan said that several banks had informed the FIC about more than 70 suspect transactions totalling some 6.8 billion rand ($500 million/440 million euros).
Gordhan, who has been a vocal opponent of corruption and excessive government spending and has repeatedly clashed with Zuma loyalists, has himself been summoned to court on fraud charges.
He says the case against him is politically motivated.
Report delayed
Sunday's criminal complaint was filed two days after South Africa's Public Protector Thuli Madonsela had been due to release a potentially-explosive report into allegations that Zuma allowed the Guptas to have undue influence over the government.
But at the last minute, the report's release was postponed following court action by both Zuma and another minister implicated in the investigation.
Madonsela's seven-year term in office as South Africa's anti-corruption watchdog ended on Saturday and the delayed report will now only be released on November 1 by new Public Protector Busisiwe Mkhwebane.
The Gupta brothers, Atul, Ajay and Rajesh, have over the years been embroiled in scandals, but the alleged extent of their political influence only came to light early this year after a deputy minister said they offered him the post of finance minister before Zuma had removed the incumbent.
