Cruyff, who made his name as a forward with Ajax and Barcelona, was European footballer of the year three times.
He won three consecutive European Cups with Ajax from 1971, coached Barcelona to their first European Cup triumph in 1992 and helped the Dutch reach the 1974 World Cup final, where they lost 2-1 to West Germany.
The Dutch FA said: "Words can hardly be found for this huge loss."
It added that Cruyff was the "greatest Dutch footballer of all time and one of the world's best ever" and wished everyone "a lot of strength in this difficult time".
Holland's friendly against France on Friday will stop after 14 minutes for a minute-long silence in tribute to Cruyff, who wore number 14 as a player.
Barcelona FC expressed "its pain and sorrow" at the news of his death, adding: "We will always love you, Johan. Rest in peace."
Gary Lineker, a former Barcelona player, said: "Football has lost a man who did more to make the beautiful game beautiful than anyone in history."
Ajax (as a player): European Cup x 3, European Super Cup x 2, Dutch league title x 8, Dutch Cup x 5, Intercontinental Cup x 1, Super Cup x 1.
Ajax (as a manager): European Cup Winners' Cup x 1, Dutch Cup x 2.
Barcelona (as a player): Spanish league title x 1, Spanish Cup x 1.
Barcelona (as a manager): European Cup x 1, European Cup Winners' Cup x 1, European Super Cup x 1, Spanish league title x 4, Spanish Cup x 1, Spanish Super Cup x 3.
Feyenoord (as a player): Dutch league title x 1, Dutch Cup x 1.
Cruyff scored 293 goals in 521 appearances for five different clubs - including 204 in 276 games while winning 18 trophies in two spells for Ajax.
He first showcased the 'Cruyff turn' at the 1974 World Cup in a match against Sweden and scored 33 goals in 48 internationals.
Cruyff had double heart bypass surgery in 1991 and gave up smoking immediately after the operation, swapping cigarettes for lollipops.
He even featured in a Catalan health department advert, warning the public: "Football has given me everything in life, tobacco almost took it all away."
Barcelona won four consecutive La Liga titles - 1991 to 1994 - under the guidance of Cruyff, who remained influential at the club after his sacking in 1995.
Cruyff was diagnosed with lung cancer in October 2015 but in February said he was "2-0 up" against his illness and was "sure I will end up winning".
Manchester City and Belgium captain Vincent Kompany called Cruyff "true football royalty", adding: "I don't think anyone has ever influenced the game as much as he has done. Football will miss him, but we will never forget."
World Cup-winning Germany captain Lothar Matthaus said Cruyff was "a great man who transformed football", while Manchester United manager Louis van Gaal called his fellow Dutchman "one of the true legends of the game".
Former Germany World Cup winner and manager Jurgen Klinsmann said Cruyff was "an amazing player, coach, teacher and person", while Manchester United and England legend Bobby Charlton said Cruyff was "one of those great, great footballers that made you excited whenever he got the ball".
England boss Roy Hodgson added: "Sometimes the word legend is used a little bit loosely, sometimes even flippantly. But there are one or two greats and one or two legends. Johan Cruyff is one of those."
Real Madrid president Florentino Perez said it was "a very sad day for the world of football", adding that he wanted to "transmit on behalf of Real Madrid our condolences and our love to FC Barcelona and most especially his wife and their children."
Former Fifa president Sepp Blatter said Cruyff gave football "a unique touch that some try to copy", while new Fifa boss Gianni Infantino said he "has marked football history for ever".
- "In a way, I'm probably immortal."
- "If I wanted you to understand it all, I would have explained it better."
- "Actually, I never make a mistake, because it takes a huge effort for me to make a mistake."
- "I don't believe in God. In Spain, all 22 players cross themselves. If it works, the game is always going to be a tie."
On football and his philosophy:
- "Playing football is very simple, but playing simple football is the hardest thing there is."
- "Italians can't win the game against you, but you can lose the game against the Italians."
- "Quality without results is pointless. Results without quality is boring."
- "It's better to go down with your own vision than with someone else's."
- "In my teams, the goalie is the first attacker and the striker the first defender."
- "There is only one ball, so you need to have it."
- "Why couldn't you beat a richer club? I've never seen a bag of money score a goal."
The Netherlands play France in Ajax's Amsterdam Arena in a friendly on Friday (19:45 GMT) and face England at Wembley on Tuesday (20:00).
BBC Radio 5 live will broadcast a special programme paying tribute to Cruyff at 19:00 GMT on Thursday, followed by another programme at 21:00 on Friday.
What were your personal memories of Cruyff? Leave your comments below.