Gerard Moreno's stoppage-time strike inflicted a fifth league defeat of the season on the reigning champions, leaving them 14 points behind leaders Barcelona, having played a game more.
The Champions League represents Madrid's best chance to end the campaign with another trophy, with the holders taking a 3-1 first-leg lead to France on March 6 in their last-16 tie with Paris Saint-Germain.
But Varane insists they must keep pushing for better results in the top flight, even though their title defence appears to be at an end.
"We performed better in the first half than we did in the second," the France international said. "We always have to give our best when we pull this shirt on.
"We can't give up on La Liga; we have to show the same drive. We're not looking at the points, we have to focus on the games and try to win them because that's what this badge demands of us. You can never write us off.
"We always want to win and this won't have any effect on the PSG game. Regardless of whether Neymar plays or not, it'll be tough because they'll come out all guns blazing. We're ready for it and will continue with our preparations."
Fellow centre-back Nacho Fernandez felt Madrid paid the price for a poor showing in the second half at RCDE Stadium and admits it has been a domestic campaign to forget.
"I don't think that we deserved to lose, but it wasn't one of our best performances," he said. "We did well in the first half and were disciplined, but in the second half our performance levels dropped off and we weren't as dominant.
"It's not normal. We're used to winning titles. We're still involved in a special competition, in the Champions League, and have to prepare for it and go in to it full of confidence."