Real were pushed all the way by a stubborn Valladolid side, with Thibaut Courtois making a couple of good saves to preserve a clean sheet.
"I'm left with a good taste in my mouth," Zidane said.
"It wasn't an easy night.
"The most important thing was to get the points because we're going to improve.
"We're always annoyed about something, more for losing possession in our own half than [refereeing] injustices.
"We weren't really on it in the first half.
"[Courtois] saved us last week and he did so today as well.
"That's what Thibaut is there for."
Vinicius Junior scored the only goal of the game after coming on as a second-half substitute.
"I'm happy for his goal and what he did," Zidane added.
"We made changes to have something more and that was how it was with those that came on."