Mr Piech had criticised his chief executive in an interview with the German news magazine Der Spiegel, but did not specify the issue at stake.
Mr Winterkorn has been widely tipped as VW's next chairman. Mr Piech and the Porsche family control 51% of VW.
Volkswagen is the biggest car manufacturer in Europe.
On 17 April, Volkswagen's five-member governing board gave its backing to Mr Winterkorn.
Board member Wolfgang Porsche, a cousin of Mr Piech, said he had given his "personal opinion" without clearing his remarks with other family members.
Mr Piech, 78, is a former VW chief executive. His wife Ursula has also resigned her seat on the board.
In a statement (in German), Volkswagen said that deputy board chairman Berthold Huber would serve as interim chairman.
During his eight-year tenure as chief executive, Mr Winterkorn has overhauled VW and made it one of the world's most successful carmakers, industry analysts say.
In 2014, VW was the world's second-biggest carmaker by sales, behind Toyota and ahead of GM.
Apart from Volkswagen, the group's brands include Audi, Porsche, Lamborghini, Bugatti, Bentley, Skoda and Seat.
The chairman of the carmaker Volkswagen (VW), Ferdinand Piech, has resigned after a power struggle with chief executive Martin Winterkorn.