Mourinho oversaw an opening-day win over Espanyol last Saturday and named an unchanged team featuring both Mbappe and Vinicius Junior once again.
The superstar duo almost engineered the game’s opening goal in the seventh minute, as Alex Remiro denied Mbappe from a tight angle after a lovely threaded ball from Vinicius. Those roles were reversed when the Frenchman picked out the Brazilian, but Remiro produced another fine save.
This contest was simmering nicely as the half wore on, and Sergio Gomez was inches away from giving La Real the lead with a superb first-time left-footed volley which flew just past the far post.
However, it was Mbappe who broke the deadlock in the 40th minute with another superb finish to add to his already extensive showreel. He took Federico Valverde’s ball into his stride and fired home a brilliant left-footed strike, scoring the first of surely many goals this season.
Sociedad responded just four minutes later, though, with Job Ochieng’s initial shot ricocheting into the path of Luka Sucic to fire home a surprise equaliser before half time.
Mbappe spurned a golden opportunity to re-establish the hosts’ advantage when Arda Guler played the France captain through on goal, but could only fire his effort straight at Remiro.
Not to be denied, Mbappe scored Real’s second after dispatching a right-footed strike past Remiro following a one-two with Jude Bellingham which bamboozled the visitors’ defence.
The England midfielder claimed another assist to double Real’s lead, showing excellent strength and persistence to hold off Luken Beitia’s challenge before laying the ball on a plate for Vinicius to apply a tap-in.
Mbappe then took centre stage once again, applying a sumptuous lob over Remiro from Vinicius’ equally impressive pass to seal all three points for the hosts in the Spanish capital.
Pellegrino Matarazzo’s side avoided any further damage with Brahim Diaz's strike ruled out for offside, but the 36-time champions head to the league summit regardless, extending their winning run in La Liga to five games stretching back to the tail end of last season.
Meanwhile, this defeat for Sociedad extends their winless run to nine games, after losing their first two in this campaign to leave them bottom of the table.
Flashscore Man of the Match: Kylian Mbappe (Real Madrid)