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.

Root, 25, made his Notts debut in 2016 and his breakthrough came this season when he made 107 from 93 balls against Warwickshire in the One-Day Cup.
Bowler Luke Fletcher has also signed a new two-year deal at Trent Bridge.
Fletcher, 28, was struck on the head while bowling at Edgbaston in July - an injury that ended his season.
Before the incident, he took 36 wickets at 22.44 to help Notts move clear at the top of County Championship Division Two.
"It's a well-deserved contract for Luke because of what he has done in the last 12 months," said Notts head coach Peter Moores.
"He's gone away and looked quite hard at what he can do to get himself into the team.
"We have seen the results of that in the way he bowled in the first part of the County Championship."
Root came through the Yorkshire Academy before being released in 2011 and joining the MCC Young Cricketers.
He is now a regular in white-ball cricket for Notts and Moores wants him to push for a place in the four-day team.
"He's been an effective player for us, particularly in one-day cricket," Moores continued.
"We're looking for him to now push on in both white and red-ball cricket after working very hard for this opportunity."

1 July 2015 Last updated at 16:19 BST
Mark Sampson's squad are only the third England team to reach a World Cup semi-final, after the 1990 men's side and the 1966 World Cup winners.
The winner of the match will face USA in Sunday's final
Sue Smith, a former England player who is commentating on tonight's semi-final, told Newsround that the World Cup had been a 'game changer' for women's football.
The chairman of the carmaker Volkswagen (VW), Ferdinand Piech, has resigned after a power struggle with chief executive Martin Winterkorn.
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Billy Root, the younger brother of England captain Joe, has signed his first professional contract, agreeing a two-year deal with Nottinghamshire.
[NEXT_CONCEPT]
England women's football team will take on Japan on Thursday in the semi-finals of the World Cup in Canada.