A spokesman for Volkswagen, which owns Audi, confirmed he was being held.
Munich prosecutors said they had acted because of a risk that Mr Stadler might seek to suppress evidence.
The scandal erupted 3 years ago, when it emerged that cars had been fitted with devices designed to cheat emissions tests.
The devices were initially found in VW's cars, but its Audi subsidiary has also been embroiled in the scandal.
Last month, it admitted that another 60,000 A6 and A7 models with diesel engines have emission software issues.
That is on top of the 850,000 recalled last year by Audi, of which only some have been found to require modification.
Munich prosecutors said Mr Stadler would be questioned by Wednesday, once he had spoken to his lawyers.
It is reported that Sueddeutsche Zeitung, VW's supervisory board has nominated Bram Schot to be the interim chief executive at Audi following the arrest of Mr Stadler.