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.
Mbappe reached five goals for the season in seven appearances across all competitions by opening the scoring when he raced into space at the start of the second half.
"I know I can do more, each game I feel better and now I'm scoring goals, and I'm happy here," Mbappe told Movistar.
"(The game was) hard, but it's the Champions League - it's always difficult, but we won at home, we won the first game," he continued.
"What we know is that the Champions League has changed and it's important to win quickly to see if we can qualify quickly or not."
The former Paris Saint-Germain striker was not able to win the club game's biggest prize while playing in France and his desire to lift the trophy matches Madrid's obsession with it.
Stuttgart did not get the memo and after Vinicius crashed an effort against the crossbar, the German side levelled.
Jamie Leweling came close himself with a flurry of attempts before hooking a corner back into the danger area for Undav to head past Courtois.
Ancelotti sent 18-year-old Brazil striker Endrick on for the final 10 minutes, hunting a winner to get the holders' trophy defence off to the perfect start.
The goal arrived swiftly but not from Madrid's galaxy of attacking stars, but instead from centre-back Rudiger, who nodded home from Luka Modric's corner in the 83rd minute.
Vinicius missed a golden chance to seal Madrid's win but his compatriot Endrick made no mistake with his only opportunity, drilling home from the edge of the box at the death.
The forward opted not to pass to his more experienced teammates and took his shot early, catching out goalkeeper Nubel.
"He's a kid with a lot of quality and he showed it," added Courtois.
"Obviously he showed personality by not passing to Vini or Kylian and shooting there himself."
