Video report by ITV News reporter Chris Skudder
Peters, whose death was announced by West Ham, was part of the Hammers trio of captain Bobby Moore and Geoff Hurst, who scored a hat-trick in the 4-2 victory at Wembley, in Sir Alf Ramsey's side.
Renowned as being the compete midfielder, Peters was good in the air, able to cross with either foot and possessed great movement, being able to drift into goalscoring positions unnoticed by his markers which would later see him nicknamed 'The Ghost'.
Peters came through the West Ham academy, having signed as an apprentice in 1959 and went on to help the east London club win the European Cup Winners' Cup in 1965.
A statement from Peters' family read: "It is with profound sadness that we announce that Martin passed away peacefully in his sleep at 4.00am this morning (Saturday).
"A beloved husband, dad and grandad, and a kind, gentle and private man, we are devastated by his loss but so very proud of all that he achieved and comforted by the many happy memories we shared. We will be making no further comment and kindly ask that the privacy of our family is respected at this extremely difficult time."
Sir Geoff Hurst, Peters' West Ham team-mate and fellow goalscorer in the World Cup final against West Germany, mourned the loss of his "partner for years".
He tweeted: "Today is a very sad day for football and for me personally. Martin Peters was one of the all-time greats and a close friend and colleague of mine for in excess of 50 years. A fellow World cup final goalscorer and my West Ham partner for years along with Bobby Moore. RIP old friend."
In 2016, it was revealed Peters had been diagnosed with Alzheimer's disease, one of several of the 1966 World Cup team to be battling dementia.
Former England and Tottenham player, Gary Lineker, paid tribute to Peters on Twitter.
He posted: "Sorry to hear that Martin Peters has passed away. One of our World Cup winning heroes. A great player and a true gentleman."
A tweet from the official England account read: "We're deeply saddened to learn of the passing of Martin Peters MBE, scorer of the Three Lions' second goal in the 1966 World Cup final, at the age of 76. Our thoughts are with Martin's family and friends at this difficult time."
After more than 300 games and some 100 goals for the Irons, Peters left Upton Park for Tottenham in 1970, becoming Britain's first £200,000 footballer in a deal which saw striker Jimmy Greaves head in the opposite direction.
While at White Hart Lane, Peters helped Bill Nicholson's team lift the League Cup at Wembley in 1971 and again, as captain, in 1973, a season after playing his part in Spurs' UEFA Cup victory over Wolves.
Spurs said they were "extremely saddened" to hear of Peters' death, tweeting: "The thoughts of everyone at the club are with his family and friends at this difficult time."
Peters, who earned 67 caps, scoring 20 international goals, also had spells at Norwich and Sheffield United before retiring in 1981.
West Ham said: "Everyone at West Ham United is deeply saddened by the tragic news that legendary Hammer Martin Peters MBE has died at the age of 76.
"The directors, management, players and staff wish to pass on our sincere condolences at this extremely sad time to Martin's wife Kath, his children Leeann and Grant, granddaughters Hannah and Meg, and all of Martin's family and friends."
Prime minister Boris Johnson also tweeted his condolences following Peters' death.
He said: "Very sad news. No England fan will ever forget the heroics of Martin Peters and his fantastic team-mates. My sympathies go out to all of those who knew and loved him."