Jonathan Silva fired Leganes into the lead on the stroke of halftime before Karim Benzema's 27th goal of the season in all competitions hauled Real level early in the second period, PA reported.
It was another lackluster display from Zidane's side, which has little left to play for this season, save for a fight with city rival Atletico Madrid for second place, after being knocked out of the Champions League and Copa del Rey.
"I as the coach and they as the players have to try," Zidane told a press conference. "We have to try and perform better because it's our duty.
"We didn't have a good first half, but improved after the break. We all have to give a bit more."
Benzema has scored seven goals in Real's last six league games and Zidane paid tribute to the France striker as speculation mounts over his future.
"Benzema is having a very good season, particularly as far as goals are concerned," Zidane said. "He's a fantastic player and this season he's scoring far more goals. I'm delighted for him.
"He's used to rumors, but that won't change anything for him. He's shown his character and he's doing a good job for the team."
Gareth Bale was introduced as an 81st-minute substitute and made little impact.
"Bale is an important player in this squad," added Zidane, who gave the Wales forward no assurances over his future last week after he had been booed off at the end of Real's narrow win over Eibar.
"I wanted to make changes today and that's it. It's difficult to get into a game when you only play 10 or 15 minutes, but substitutions are made to change the game and we were unable to do that."
Leganes is unbeaten in its last four games and has moved up three places to 11th in the table, but boss Mauricio Pellegrino is not yet satisfied.
"We will try to do our best until the end of the season," he told a press conference. "It's not the same for the club to go 10th and 16th, we'll go for more.
"The team has been competing well, better and better. We are going from less to more and it is good news for the family of peppers."