European champions Real lost 2-1 in the second leg at their Bernabeu home as the tie finished 2-2 on aggregate with their modest city neighbours going through on away goals despite having lost the first leg 1-0 last week.
"I am responsible for all of this. It's a failure for me," said Zidane who recently penned a contract extension until 2020.
Admitting that it had been his worst night as a coach, the Frenchman added: "It's a logical result, our opponents played their match and we did not.
"I am the manager. I picked the team and I was wrong in many regards. It's up to me to find the solutions. I will continue to fight, to work, to battle and to find the things the team needs."
Javi Eraso gave Leganes a 32nd-minute lead with a spectacular curling shot from long range.
Karim Benzema levelled on the night after 47 minutes thanks to a pass from Lucas Vazquez but Brazilian striker Gabriel Pires then hit the winner with a firm header eight minutes later.
The fans let their feelings known by booing off the team at half-time.
Leganes have a budget of just 45 million euros -- 15 times less than the 675 million euros commanded by mighty Real.
Even without Cristiano Ronaldo and Gareth Bale, who were rested after starring in the 7-1 rout of Deportivo La Coruna at the weekend, Real should still have had enough firepower for a side languishing in 13th spot in La Liga.
But Leganes put on an inspired display, despite having their backs to the wall late in the game, to make the semi- finals for the first time.
In the night's other game, Valencia made the semi-finals but needed a 3-2 penalty shoot-out to defeat Alaves who won 2 -1 for a 3-3 aggregate.
Yesterday, Sergio Escudero's goal after just 24 seconds set Sevilla on their way to a 3-1 win over Atletico Madrid and a place in the semi-finals. Sevilla, finalists in 2016, had won the first leg 2-1.
Coutinho, who has been handed the number 14 shirt vacated by China-bound Javier Mascherano, had been sidelined by a thigh injury since his drawn-out transfer saga came to an end on January 6.
The 14 shirt was also worn by the club's Dutch legend Johan Cruyff who died in 2016.
Coutinho has been cleared to make his debut at Camp Nou as Barcelona look to overturn a 1-0 defeat to Espanyol in the first leg of their quarter-final clash.
Club captain Andres Iniesta and striker Paco Alcacer were also included in the 19-man squad for Thursday's game as was Colombian defender Yerry Mina, another transfer window recruit.
Disclaimer: No Business Standard Journalist was involved in creation of this content
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