BERLIN— Volkswagen AG raced Wednesday to contain the widening scandal threatening Germany’s most important company, ousting its chief executive and pledging to prosecute those involved in a scheme to cheat U.S. auto-pollution tests.

CEO Martin Winterkorn’s resignation follows a calamitous few days after Friday’s disclosure by the U.S. Environmental Protection Agency that Europe’s biggest auto maker employed software on some VW and Audi diesel-powered cars to manipulate the results of routine emissions tests.

The crisis threatens to spill beyond the auto maker to the broader German economy. Wolfsburg-based Volkswagen is as much institution as corporation at home, with nearly 300,000 employees, 29 plants across the country and deep ties to the government—Lower Saxony owns 20% of VW.

The company’s next CEO faces a daunting task of cleaning up the scandal—the scope of which remains unclear—and keeping its sales expansion on track. Volkswagen hasn’t yet said it knows who was responsible or how many employees were involved.

On Tuesday, Volkswagen disclosed that as many as 11 million cars contained software alleged to have duped emissions tests and were possibly subject to a global recall. The company issued a profit warning and disclosed a €6.5 billion ($7.27 billion) charge to earnings to cover the costs of addressing the matter.

In a statement following Wednesday’s meeting of the company’s top shareholders and labor representatives, Mr. Winterkorn said he would “accept responsibility” for the “irregularities that have been found in diesel engines” and tendered his resignation to the supervisory board.

“I am shocked by the events of the past few days,” he said. “Above all, I am stunned that misconduct on such a scale was possible in the Volkswagen Group.”

The executive committee of the supervisory board thanked Mr. Winterkorn for his contributions to the company and said the CEO had “no knowledge of manipulation of the emissions data.”

The committee said it would seek prosecution of any Volkswagen employees involved in the affair, and it would establish a special investigative committee to uncover what had happened and who was responsible.

The board subcommittee said it would present by Friday’s scheduled supervisory board meeting names of candidates to succeed Mr. Winterkorn, but didn’t disclose any.

Two prominent Volkswagen executives on many lists are Herbert Diess and Matthias Müller.

Mr. Diess is a former BMW AG executive who joined Volkswagen in July after being passed over at BMW for the CEO’s job. He runs its namesake brand, the company’s biggest business. Choosing Mr. Diess would send a signal that Volkswagen shareholders are opting for a fresh start, bringing in an outsider with a strong record in cost-cutting.

Some analysts, however, say Volkswagen has never done well with outside executives. The company’s culture is famously clubby and success depends on being well-connected in Wolfsburg and striking a balance between boosting profit margins and maintaining strong ties to labor.

Others think that insular culture makes Mr. Müller, CEO of sports car brand Porsche, a more likely candidate. Mr. Müller gets along well with the Porsche and Piech families who control the company, said people familiar with the matter, and he has deep roots in Volkswagen. He is also well respected in financial circles. “We believe shareholders would welcome such a move,” said Arndt Ellinghorst, automotive analyst at Evercore ISI.

Whoever becomes the next CEO, the job of boosting profit margins likely will take a back seat to steering the company through what could be years of rebuilding a brand badly wounded over the past few days. The company’s market value is off 29% since Friday.

Volkswagen could face more than $18 billion in fines from the EPA, though analysts say it is unlikely that Volkswagen will have to pay that much. The U.S. Department of Justice has launched a criminal investigation that could result in indictments against Volkswagen executives, analysts said.

The crisis is spreading as regulators and justice officials in Europe and Asia launch investigations, and angry investors and customers file lawsuits seeking damages.

The urgency to repair Volkswagen’s reputation goes beyond the benefits to shareholders or even the company’s 600,000 employees world-wide. Volkswagen is Germany’s largest corporation, generating revenue of almost €203 billion ($227 billion) last year in a country where every seventh job is linked to the nation’s export-oriented auto industry.

Damage to Volkswagen could prove a major blow to the broader German economy. Some German politicians, though angry at the company for violating U.S. law, say Volkswagen is being singled out by U.S. authorities.

“It’s no coincidence that this discussion comes up now,” said Oliver Wittke, a conservative German lawmaker. “Economic interests in the U.S. are also playing a role here.”

Under Germany’s two-tier corporate governance, the supervisory board oversees the executive management board, but doesn’t run the company day-to-day. A five-person subcommittee of the supervisory board, including Wolfgang Porsche, whose grandfather was Beetle inventor Ferdinand Porsche, gathered in Wolfsburg for a crisis meeting. The Porsche-Piech families control about 51% of Volkswagen’s voting stock. The second-largest shareholder is the state of Lower Saxony, which holds 20% of the voting rights and has special privileges. State Premier Stephan Weil was present at the meeting. Also in attendance were three representatives of the company’s workforce and IG Metall trade union, including Bernd Osterloh, the powerful head of its works council.

ENLARGE

It is still unclear who at Volkswagen was responsible for the scheme to trick the EPA. The company hasn’t offered a rationale, though outside experts speculate it was to ensure strong engine performance and boost fuel economy amid tough U.S. emissions standards.

The members of the executive committee meeting in Wolfsburg said they were convinced that Mr. Winterkorn knew nothing about it. VW is launching an investigation of its own and will tap external experts. It also asked prosecutors in Braunschweig, the county where VW is located, to investigate.

“I’m pleased that Volkswagen is taking such an aggressive stance on admitting the problem and attacking it,” Gina McCarthy, EPA administrator, told The Wall Street Journal on Tuesday.

Mr. Winterkorn’s resignation is a bitter end to a long career at Europe’s biggest car maker. He was hired by its luxury car brand, Audi, in 1981 as assistant to the director of quality control. A stickler for detail, he became known for his obsession with the quality of its vehicles. He became CEO of the company in 2007 and oversaw a period of unparalleled expansion. He aimed to make Volkswagen the biggest, most profitable and best-run car company in the world. It could overtake current market leader Toyota Motor Corp. in annual sales. In the first half of this year, Volkswagen sold more cars than its rival. But profits are falling as higher costs hit margins, and sales in markets such as China and the U.S. decline. Now, he said, it is time to step down. “Volkswagen needs a fresh start,” he said. “I have always been driven by my desire to serve this company, especially our customers and employees. Volkswagen has been, is and will always be my life.”