The prosecutors did not name the manager, but according to news site Spiegel Online, he is Giovanni Pamio, who led a team of engineers responsible for designing emissions control systems for diesel vehicles in the United States between 2006 and late 2015.
Pamio, 60, was formally charged by the US Department of Justice in a criminal complaint in New York on Thursday with conspiracy to defraud the United States, wire fraud, and violation of the Clean Air Act.
The prosecutors in the southern German city of Munich said the suspect was arrested on Monday and was now being questioned in custody.
He is one of several people under investigation on suspicion of fraud and dishonest advertising, senior prosecutor Andrea Grape said.
But she added that no members of the Audi board were targeted in the probe.
According to the DOJ, 60-year-old Pamio directed employees to design software to cheat US emission tests.
Volkswagen, the world's largest carmaker, admitted in September 2015 to using so-called "defeat device" software to cheat regulatory nitrogen oxides emissions tests in some 11 million cars worldwide.
The devices allowed the cars to spew up to 40 times the permissible limits of nitrogen oxide during normal driving, but this was hidden during emissions testing.
VW faces an array of legal challenges in Germany and worldwide relating to the software, installed mainly in own-brand vehicles but also in cars made by Audi, Skoda and Seat, among its stable of 12 brands.
According to the US DOJ, "Pamio directed Audi employees to design and implement software functions to cheat the standard US emissions tests.
"Pamio and co-conspirators deliberately failed to disclose the software functions, and they knowingly misrepresented that the vehicles complied" with US emissions standards.
In total, Volkswagen has agreed to pay some $23 billion in the United States to compensate some 600,000 US car owners.
Volkswagen previously pleaded guilty to three felony counts connected to the cheating scandal, and in April was ordered to pay a $2.8 billion fine.
US authorities made their first indictment of a Volkswagen employee in the "dieselgate" scandal in September 2016. The accused US engineer pleaded guilty to avoid a lawsuit.
Six other company executives have since been implicated, one of whom was arrested in January in Miami.