Müller, 62, will take over with immediate effect, replacing Martin Winterkorn who stepped down two days earlier, said the head of the supervisory board, Berthold Huber, in the carmaker's headquarters in Wolfsburg, northern Germany.
"My most pressing task will be to restore confidence in the Volkswagen Group - through an unsparing investigation and maximum transparency, but also by drawing the right lessons from the current situation," Müller vowed.
"Volkswagen under my leadership will make every effort to develop the most rigorous compliance and governance standards in the entire industry and to implement them."
The emissions scandal broke a week ago when US officials publicly accused Volkswagen of cheating and launched a probe which has also seen a growing list of other countries launch investigations.
The company admitted earlier this week that 11 million of its diesel vehicles worldwide are equipped with the software that covertly turns on pollution controls when the car is being tested, and off when it is being driven.
German Transport Minister Alexander Dobrindt told the Bundestag (German parliament) on Friday that 2.8 million cars in Germany were caught up in the scandal.
The number of cars affected by the cheating in Germany dwarfs the 500,000-odd vehicles believed to have had the software installed in the US.
'A small group of people' to blame
Huber praised Müller as "a figure with great strategic, entrepreneurial and social skills" as he is set to dive head-first into quelling the storm of the scandal.
"He knows the company and its brands, will tackle his new position immediately and with full force. We explicitly appreciate his critical and constructive views."
The supervisory board also said on Friday that the decision to fit devices on cars to cheat emissions tests was a "moral and policy disaster," adding that a small group of people was to blame.
"We can clearly say that emissions manipulation was a moral and policy disaster," said Huber, who heads the committee of 20. Another board member, Bernd Osterloh, added that "a small group had caused great damage to Volkswagen."