The Colombia forward has joined the European champions from Monaco on a six-year deal believed to be worth €80 million on the back of an outstanding World Cup.
Rodriguez is the Spanish giants' second close-season new recruit following the arrival of Toni Kroos from Bayern Munich and the 23-year-old spoke of his determination to add to the club's long list of honours.
"I'm very happy to be here, it's a dream becoming a reality for me. I want to give you lots of happiness and titles in the future," he said at his presentation ceremony.
Rodriguez inspired his country's impressive run to the quarter-finals of the World Cup and won the Golden Boot by scoring six times - including the goal of the tournament against Uruguay.
A huge crowd flocked to the Bernabeu to see Rodriguez introduced as a Real player.
Real president Florentino Perez believes the arrival of the former Monaco man is another statement of the club's intent to follow up their Champions League and Copa del Rey triumphs last season by winning yet more trophies.
Perez: "We've signed a player that has led a fantastic Colombia national team in the recent World Cup. A player that is one of the stars of the World Cup. His goals have shown that, including the goal of the tournament.
"Last season was magnificent, but we can't sit back on that and be complacent."
Rodriguez spent just one season at Monaco after his move from Porto for a fee in the region of €45million in May last year.
He quickly justified that outlay, scoring 10 goals and being named in Ligue 1's team of the season as Monaco finished second behind Paris Saint-Germain.
