However, Ancelotti praised his players' resilience and was proud of how they were able to endure the home side's pressure and claim three valuable points.
"It was a complicated match. We didn't deserve to win, because Real Sociedad pushed us really hard," he told a press conference.
"We held on and suffered, showed character and a lot of commitment. I value that a lot because it's not easy to find commitment in a team with so much talent and quality and today we did it.
"I leave very satisfied and then we have to be self-critical in order to improve. It's a great opportunity for us to assess what we didn't do well, but with three more points in the table."
Real remain second but now have 11 points, one behind Barcelona who visit fifth-placed Girona on Sunday.
Ancelotti acknowledged his side were struggling with fatigue and poor form, but said it was to be expected due to a busy calendar and coming off the back of the international break.
The Italian said he saw improvement in Mbappe's performance as the France captain settles in at his new club, having scored his third goal for Real on Saturday.
"We can talk about all the problems in the world with this match, but the important thing was to get out with a win," added Ancelotti, who was without midfielders Jude Bellingham, Eduardo Camavinga, Dani Ceballos and Aurelien Tchouameni.
"We have to take into account that we are not at 100 per cent. It's normal, we don't have four midfielders who are injured. I value a lot the work of those who have played.
"I see Mbappe fresher, more active. He's very dangerous and combines well with Vinicius and the other forwards. He is improving a lot. I liked his game a lot."
Real will now turn their attention to the Champions League title defence starting on Tuesday when they host Stuttgart.