Antonio Rudiger headed home in the 83rd minute for the record 15-time champions after Deniz Undav had levelled Mbappe's opener just seconds into the second-half.
Endrick, another Champions League debutant, wrapped up Madrid's win with a low drive in stoppage time.
Stuttgart had the better of an entertaining opening period but Carlo Ancelotti's team have a habit of grinding out victories when up against the wall and they took the three points despite the visitors' impressive efforts.
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Comience su prueba gratuita de Magzter GOLD de 7 días para acceder a miles de historias premium seleccionadas y a más de 9,000 revistas y periódicos.
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