Heinze, 29, has signed a four-year deal with Real while the 23-year-old Robben will be tied to the Spanish giants for the next five seasons.
The two players undertook their medicals with the Spanish champions and, after putting pen to paper on their new contracts, were unveiled in front of fans and media at the Bernabeu.
Neither fee has been confirmed, but reports in the Spanish media suggest the double swoop could cost Real almost 50million euros (£34million) in total.
Heinze admitted replacing the Brazilian Roberto Carlos will be a tough task.
Heinz: "I think that it is impossible to forget Roberto Carlos. I have come here to give everything. What Roberto Carlos has done is impressive.
"He (Schuster) will know where I can be most useful. At Manchester United I always played at left-back and for Argentina in the middle, apart from the Copa America."