Madrid dominated the fixture from start to finish, produced 16 shots and five big chances, with only an excellent performance from Sociedad keeper Unai Marrero preventing them from scoring more.
Real broke the deadlock in the 38th minute after being awarded a penalty for a foul on Arda Guler in the box.
Mbappe missed the spot kick but was on hand to tuck away the rebound to hand his side a well-deserved lead.
Despite their dominance, Los Blancos had to wait until the 83rd minute to secure the points when Mbappe added his second goal of the game.
Elsewhere in LaLiga on Saturday, Valencia and Real Betis traded goals in a thrilling 1-1 draw, while Espanyol proved to be too strong for Las Palmas in a 2-0 win.
Leganes concluded the season with a bang with a 3-0 demolition of Real Valladolid, with Alaves and Osasuna playing out to a 1-1 stalemate.
Getafe went down 2-1 to a hard-working Celta Vigo outfit, and Rayo Vallecano and Mallorca cancelled each other out in a drab 0-0 draw.