The Ligue 1 side were unable to resist the bumper profit on the 23-year-old, who had been signed for €45 million (£35m) last summer, and the transfer was completed on Tuesday.
James starred at World Cup 2014, winning the Golden Boot and Goal of the Tournament while narrowly missing out on the Golden Ball award to Argentina's Lionel Messi.
His performances as Colombia reached the quarter-final convinced European champions Real Madrid to splash the cash, but the club's vice-chairman stressed that Monaco's ambition has not wavered despite the departure setback.
"Monaco are proud to have helped James Rodriguez reveal his talent to the world as he notably made his mark on the 2014 World Cup in Brazil winning the Golden Boot," Vasilyev told the Stade Louis II side's official website.
"James is a fantastic player who achieved great things with Monaco and the club had no intention to sell the player.
"However the time came when the solution of a transfer was considered to be the most beneficial solution for all parties. Monaco are proud to be a part to one of the most significant transfers in football history.
"Monaco are a club that will continue to grow and remain focused on the future. More than ever we have the ambition to strengthen our team and continue to progress for next season."
The principality club finished second in Ligue 1 last season and will compete in the Champions League in 2014-15, albeit without Madrid-bound James.