Madrid won 2-0 via penalties from Vinicius Jr and Kylian Mbappe.
Ancelotti later said: "It was a complicated game and we probably didn't deserve to win because Real Sociedad played very well, but we held on, we suffered and we were committed, which I value very highly. You can't always find sacrifice in a team with so much quality and today we did.
"We are not at 100%, but individually the players are improving a lot. We're very happy because the important thing was to get off to a good start in this run of matches, of which there are many. When you're missing four midfielders and we're not at 100% it's very difficult to find the balance. The work of the midfielders who played today, Modrić, Valverde and Arda Güler was very good, something I value very much.
"I come away very satisfied overall, but we have to make the necessary criticism in order to improve. This match is a great opportunity for us to critically assess what didn't go well, but we picked up three more points in the standings, which is the most important thing."
He added, "I didn't see the second penalty. When the referee went to the VAR he saw it better and I don't think he had any doubts when it came to awarding it. The first one was scored by Vini Jr. and the second one they decided that Mbappé should take. They share the responsibility of taking them and they're doing a good job."