The Colombia international signed a six-year deal with the UEFA Champions League winners on Tuesday after starring at the FIFA World Cup, winning the Golden Boot following his six-goal haul.
Real are believed to have paid around €80million for the playmaker, who spent just one season in Ligue 1 after joining Monaco from Porto.
Vasilyev said the deal was too good for all parties to ignore and believes his club will continue to improve, even without Rodriguez.
"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," he told the club's 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."