Los Blancos had not won in the three matches prior to the quarter-final first-leg at Estadio Municipal de Butarque, where Marco Asensio found the net in the 89th minute.
"It is important to get a win for the team's confidence, and we kept a clean sheet," said Varane.
"The coach told us we needed balance and the team's focus was great. We played worse than the other day but won. Other times we have played well and lost.
"We want to win and get on a good run. We are playing the same as last year but things are not coming together. We enjoy winning we love to compete and relish coming out on top."
Madrid find themselves fourth in the Spanish league standings, 19 points behind Barcelona, leaving the cup competitions as their best hopes for winning silverware.
While they will expect to finish off Leganes at the Santiago Bernabeu next week, Varane insisted that Real are not thinking about their Champions League clash with Paris Saint-Germain in February.
"We're not thinking about the PSG tie yet, we're just focused on winning games. We have many games left, we want to get better and be hard to beat like today."
