Chelsea confirmed the death of Bonetti following what it described as a "long-term illness", adding: "All at Chelsea wish to send our heartfelt and deepest condolences to Peter's family and friends."
Cech, who surpassed Bonetti's Chelsea record of 208 clean sheets in 2014, posted on Twitter: "Really sad news for everyone from @ChelseaFC family.. RIP "The Cat"... Legend."
Bonetti made 495 appearances for Chelsea, a record beaten only by Harris, who described him as the club's greatest goalkeeper, and "a great lad".
Speaking to Sky Sports News, Harris said: "He was about 5ft 10, ever so slim, and acrobatic. He pulled off some unbelievable saves.
"I'm not knocking any of the present day keepers but ask any Chelsea supporter who's been around for some time and I bet they'd say the number one keeper at Chelsea was Peter Bonetti."
John Terry, who made 492 appearances for the club, just three short of Bonetti, said he was "absolutely heartbroken" to hear the news.
Terry wrote on his Instagram account: "Absolutely heartbroken. RIP Peter Bonetti – @Chelseafc LEGEND & HERO. Sending my love and condolences to Peter's family at this terribly sad time. A true gentleman."
Bonetti, nicknamed 'The Cat' for his outstanding reflexes and agility, made his Chelsea debut as an 18-year-old, keeping a first clean sheet in March 1960.
Harris and Bonetti were long-time team-mates, as was Bobby Tambling, who scored 202 Blues goals, a record only beaten by Frank Lampard in May 2013.
His Chelsea spells were punctuated by a brief stint in the United States with St Louis Stars. He also had spells with Dundee United and Woking.
Bonetti made seven appearances for England, including in the 1970 World Cup finals, but Gordon Banks was always preferred.
He was in the 1966 World Cup-winning squad, but did not play, belatedly receiving a winners' medal in June 2009.
Bonetti was twice promoted to the First Division with Chelsea and won the 1964-65 League Cup, 1970 FA Cup and the UEFA Cup Winners' Cup in 1971.
Peter Shilton, who was part of England's 1970 World Cup squad alongside Bonetti, tweeted his respects, describing Bonetti as "a hero of mine, a tremendous player and a true gentleman."
FA chairman Greg Clarke said he was "very saddened" to hear of the death of "a wonderful goalkeeper but also a hugely-popular figure in the English game."
After retiring as a player, Bonetti took up coaching roles at Newcastle, Fulham and Manchester City, whose former midfielder Joey Barton paid tribute.
Barton wrote on Twitter: "Really saddened to hear of the passing of Peter Bonetti. He was our goalie coach @ManCity under KK. All round top bloke and gentlemen. RIP Cat. Broken heart. Thoughts are with his family and friends."