Tilkowski died in the company of his family on Sunday after a long illness, the club said.
Tilkowski won 39 caps for West Germany between 1957 and 1967, with his name forever linked with one of the most famous goals in football history.
It came at the 1966 World Cup final against England at Wembley when England striker Geoff Hurst beat Tilkowski with a shot in extra-time which hit the underside of the bar and span away from goal after hitting the ground.
The goal was awarded, after Swiss referee Gottfried Dienst consulted then Soviet linesman Tofik Bahramov, to give England a 3-2 lead before Hurst hit a late fourth, but the debate on whether or not the ball crossed the line has continued ever since.
Hurst, now 78, said on Twitter he was "very touched" that Tilkowski's family had called to inform him of the goalkeeper's death.
"Very sad to receive a call earlier to let me know that Germany's goalkeeper from 66 World Cup, Hans Tilkowski, has died," he said.
"Terrific player for his club, Borussia Dortmund, and country and a very fine man, I very much enjoyed the time we spent together over the years."
Tilkowski was also a German Cup winner with Dortmund in 1965 and helped Dortmund become the first German team to win a European title when they beat Liverpool 2-1 at Hampden Park, Glasgow to win the final of the 1966 European Cup Winners' Cup.
He later played for Eintracht Frankfurt, spent more than a decade as a coach and was in later life active in charity work, raising more than 1 million euros for the United Nations's Children's Fund UNICEF and for cystic fibrosis and multiple sclerosis charities.
Dortmund president Reinhard Rauball said German football had lost "an internationally well-respected athlete" who "wrote football history" with the club.
"Tilkowski's saves in the final (against Liverpool) and on the way to the final played a crucial part," he said.
Tilkowski was in 1965 the first German goalkeeper to be voted his country's footballer of the year, and was regarded as one of the best goalkeepers of an era which also included Soviet great Lev Yashin and England's Gordon Banks.
Banks, who was on the opposite side of the field to Tilkowski in the Wembley World Cup final, died last February at the age of 81.
German football federation DFB president Fritz Keller said, Tilkowski was one of the best goalkeepers in the world at the time.
"The World Cup title would have been the well-deserved crowning of his career in the national team," Keller said.
But despite being on the losing side in the 1966 final as a result of the controversial Hurst goal, Tilkowski was against goal-line technology, which was finally introduced at a World Cup in 2014.
"I am against it because it will lose the attractiveness of football and the discussion about football," he said a few days before his 80th birthday. "Who will be able to talk about a goal like Wembley's for years to come?"