VW admitted in September 2015 to using illegal software to cheat U.S. emissions tests on diesel engines, sparking the biggest crisis in the company's history and leading to a regulatory crackdown across the auto industry.
The United States filed criminal charges against former VW CEO Martin Winterkorn in May, but he is unlikely to face U.S. authorities because Germany does not extradite its nationals to countries outside the European Union.
The Munich prosecutors said Stadler's arrest was not made at the behest of U.S. authorities. The executive was arrested at his home in Ingolstadt in the early hours on Monday, they said.
"As part of an investigation into diesel affairs and Audi engines, the Munich prosecutor's office executed an arrest warrant against Mr Professor Rupert Stadler on June 18, 2018," the prosecutor's office said in a statement.
A judge in Germany has ordered that Stadler be remanded in custody, it said, to prevent him from obstructing or hindering the diesel investigation.
Audi and VW confirmed the arrest and reiterated Stadler was presumed innocent unless proved otherwise. Stadler himself was not immediately available for comment.
At 1040 GMT, VW shares were down 2.7 percent at 156.70 euros, one of the biggest falls by a European blue-chip stock .FTEU3 .
Stadler has been under fire since Audi admitted in November 2015 - two months after parent VW - that it also installed illegal "defeat device" software to cheat U.S. emissions tests.
Munich prosecutors are investigating whether Stadler acted swiftly enough to stop deliveries of manipulated Audi models in Europe once emissions problems had emerged, a person familiar with the matter has told Reuters.
Stadler has held onto his post mainly thanks to the backing of members of VW's controlling Porsche-Piech families. Before becoming Audi CEO in 2007, Stadler worked as chief of staff to VW's former chairman Ferdinand Piech.
Earlier this month, Munich prosecutors widened their probe at Audi to include Stadler and another member of Audi's top management, investigating them for suspected fraud and false advertising.
Stadler's arrest will raise tensions on VW's supervisory board, putting at risk a fragile truce between management, labour representatives and board members from the carmaker's home region of Lower Saxony.
A spokesman for Porsche SE, the company that controls VW and Audi, said Stadler's arrest would be discussed at supervisory board meetings of both VW and Audi on Monday.
VW has so far set aside around $30 billion to cover the cost of fines, vehicle refits and lawsuits related to its emissions test cheating.
Most of its problems have been in the United States, where a total of nine people have been charged and two former VW executives have pleaded guilty and been sentenced to prison terms.
But investigations are continuing elsewhere. Last week, German prosecutors fined VW 1 billion euros ($1.2 billion) over the scandal.
($1 = 0.8608 euros)
(Reporting by Jan Schwartz, Ilona Wissenbach, Edward Taylor, Joern Poltz and Irene Preisinger; Editing by Maria Sheahan and Mark Potter) ((Edward.Taylor@thomsonreuters.com; +49 69 7565 1187;))