The French forward, who joined Madrid this summer, scored his third goal in two matches, calmly converting a penalty late in the second half to secure all three points for Los Blancos.
Mbappe, who opened his La Liga account before the international break, stepped up after Vinicius Jr. won a penalty when Jon Aramburu accidentally stood on his foot. The Frenchman showed no nerves as he sent Sociedad goalkeeper Alex Remiro the wrong way, giving Madrid a 2-1 lead in the 82nd minute.
Vinicius Jr. had already opened the scoring in the 58th minute, also from the penalty spot, after Sergio Gomez, the former Manchester City left-back, was penalised for handling Arda Guler’s fierce shot in the box. The Brazilian made no mistake, dispatching the penalty with confidence to score his second La Liga goal of the season.
Real Sociedad, however, might feel hard done by after dominating large portions of the game and striking the woodwork three times. Luka Susic, the Croatian sensation, was the first to hit the crossbar with a powerful shot from outside the box in the 25th minute, leaving Real Madrid keeper Thibaut Courtois stranded.
Ten minutes later, Sheraldo Becker's left-footed strike also hit the bar after Dani Carvajal lost possession deep in Madrid's half.
Susic’s bad luck continued just 20 seconds into the second half when he managed to get past the Madrid defense, but his effort hit the post, denying Sociedad an early second-half lead. Despite their persistence, Sociedad could not convert their chances into goals.
Real Madrid, missing England international Jude Bellingham due to a calf injury, relied on their defensive resilience and clinical finishing to grind out the victory. Bellingham, who was injured before the international break, has returned to training but was not risked for this match.
With the win, Real Madrid maintain their unbeaten run after five league matches but remain a point behind league leaders Barcelona, who have played one game less. The victory keeps Los Blancos in close pursuit of their rivals, with Mbappe and Vinicius forming a dangerous partnership at the heart of their attack.