Cristiano Ronaldo’s 24th goal in 15 games this year was not enough to earn his side a derby victory as an Antoine Griezmann equaliser secured Diego Simeone’s side a point.
The result saw Real drop to fourth place in La Liga behind Valencia, who beat Espanyol 1-0 later in the day.
Ronaldo was taken off shortly after the hour-mark at the Santiago Bernabeu while Luka Modric and Karim Benzema both started on the bench, with Zidane keeping one eye on Wednesday’s quarter-final second leg at home to Juventus.
Real boast a 3-0 advantage from the opening leg in Turin and Simeone suggested on Sunday that Los Blancos should now be considered favourites to claim a third consecutive European triumph.
Zidane, however, is wary of complacency.
“Everyone sees us going through to the semi-finals and that’s exactly what we should not put in our head,” Zidane said. “We know very well that we will have to suffer again on Wednesday.”
Atletico, meanwhile, will now turn their attention to the Europa League and Thursday’s quarter-final second leg against Sporting Lisbon. They won the home leg 2-0 last week.
They should arrive in confident mood after retaining their four-point cushion over Real in second place while it was perhaps the perfect result for leaders Barcelona, who are now 11 points clear at the top.
“We deserved more today clearly, it’s a shame,” Zidane said. “We had many chances, but in the end we could have lost the game as well.”
Griezmann’s goal came on his 200th appearance for Atleti while Ronaldo’s was the 500th in the 161 league matches played between these two city rivals.
On Griezmann, Simeone said: “He is a great player, he did not have a good start to the season and he knows it and it is not easy to reinvent himself in the way he did in November, December.
“I think the presence of (Diego) Costa has helped him a lot, having strength around him which allows him to be free.”