Zidane's side had to come from behind at home to beat Sporting Lisbon, 2-1, in their Champions League Group F opener on Wednesday.
The defending champions needed late goals from Cristiano Ronaldo and Alvaro Morata to collect three points.
Their performance paled in comparison to the rampant display of El Clasico rivals Barcelona, who rolled to a 7-0 win over Celtic in their first group game.
Zidane said he never doubted that Real would win, telling his club's official website: "It wasn't unexpected.
"I don't know whether we deserved it or not, but we kept believing right until the end.
"At any given time we can make the difference.
"We didn't throw in the towel and that is important. The effort put in was tremendous."
On Thursday, El Mundo newspaper wrote "You can't always win at the last second or on penalties." The daily's reference was to Real's penalty shootout triumph over Atletico Madrid in the European final in May, and the 3-2 win over Sevilla in extra-time that gave Real the European Super Cup last month.
Zidane said Real's narrow victories were still important.
"When you win a game in this manner, it means a lot to the squad," the Frenchman said. "That's what I take away with me. We can't win all of our games 5-0."
Ronaldo, playing against his boyhood club, cancelled out Bruno Cesar's opener with an 89th minute free-kick before substitute Morata, who replaced Karim Benzema, headed in the winner on the last play of the game.
Real have won all of their games this season and lead the La Liga standings after three rounds but Zidane accepted improvement was needed.
"We're aware that the performance wasn't a perfect one," he said. "We can perform better than we did. We're only at the start of the season."
The Spanish media believe Zidane made a mistake by giving Benzema his first start of the season against Sporting with the French forward having only recently recovered from injury.
El Mundo labelled Benzema's performance "disastrous" and claimed "he is not even fit to play for (Real's reserves) Castilla."
Zidane, however, thought otherwise.
"He did well, despite not having played all that much. It was to be expected that he started the game a bit slowly," he said.
(Reporting by Adriana Garcia; editing by Amlan Chakraborty)
