MbappĂ© broke the deadlock in the 40th minute, controlling a driven ball from Federico Valverde before finishing into the top corner, only for Luka SuÄiÄ to level for the visitors moments before half-time.
Mourinho admitted his side had been caught out.
"Real Sociedad is a very good team that created difficulties for us in the first half. I would have liked a cooling break around minute 20 or 25 to talk to the players â they caught us by surprise. They didn't create too much danger, but it was difficult for us to press," he said.
"We changed things in the second half. It was important to stay calm, because pressing blindly and without reference points would have been more complicated. We kept that composure, and the second half was very, very strong."
MbappĂ© restored Real Madrid's lead early in the second half with a bending finish after a one-two with Jude Bellingham, before VinĂcius JĂșnior made it 3-1, tapping in after Bellingham's third assist of the night. MbappĂ© completed his hat-trick with ten minutes remaining, latching onto a pass from VinĂcius, and Brahim DĂaz had a fourth Madrid goal ruled out for offside late on.
Mourinho did not hold back in his praise for the French forward.
"There's something I don't like doing, which is talking about individuals, but what Kylian has done... it's crazy," he said. "He's incredible. What he does, his hunger... I'm very happy he made history at the World Cup as top scorer. But I insist: the team has always been the team, and that's the most important thing."
Real Sociedad head coach Pellegrino Matarazzo felt his side had lacked the quality to make the most of a promising opening period.
"Look at the goals â in terms of how we defended as a block, aside from a few moments where we sat too deep, they were precise, with a lot of quality, but we weren't good for the second goal, and for the third we lost the ball first, then lost our marker," he said.
"Maybe we were tired and lost some aggression, and we suffered a lot when we lost the ball. I think we had too much possession in our own half â we needed to be more direct and attack the box, which is how we scored our goal, but we didn't try it enough."
Matarazzo also explained his decision to withdraw Mikel Oyarzabal after 55 minutes, having managed the forward's return from injury carefully.
"We had planned for him to play the first half and 10 more minutes... when he was on the pitch we competed better. For us he's a leader, in attack and in defence," he said.
