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The 23-year-old has signed a six-year deal with the Spanish giants.
Rodriguez won the Golden Boot with six goals in five matches in Brazil as Colombia reached the quarter-finals.
Madrid president Florentino Perez said: "He played brilliantly and his goals confirmed his status as one of the best players at the World Cup."
Rodriguez's move is the fourth most expensive transfer of all time after those of new team-mates Gareth Bale and Cristiano Ronaldo and Barcelona's Luis Suarez.
"I am very happy, this is a dream come true. I hope to make people very happy and win a lot of titles here," he said.
The Colombian scored against Greece, Ivory Coast and Japan as they topped Group C before netting twice, one a sublime volley, during the 2-0 last-16 win against Uruguay, which was voted goal of the tournament.
Rodriguez's final goal was a late consolation from the penalty spot as Colombia lost 2-1 to Brazil.
The former Porto forward won the Portuguese league three years in a row before joining Monaco last season, where he scored nine goals in 34 Ligue 1 games.
Monaco vice-chairman Vadim Vasilyev said his club had no plans to sell Rodriguez, but were proud to be part of "one of the most significant transfers in football history".
Vasilyev added: "The time came when the solution of a transfer was considered to be the most beneficial solution for all parties."
Madrid made Ronaldo the world's most expensive player when they signed him for £80m from Manchester United in 2009, and eclipsed that with the £85.3m they paid Tottenham for Gareth Bale last summer.
Rodriguez's signing comes less than a week after they signed German World Cup winner Toni Kroos, who moved from Bayern Munich for an undisclosed fee.
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Real Madrid have signed Colombia forward James Rodriguez, the top scorer in the 2014 World Cup, from Monaco for a fee which could reach £71m.