Youngster Oscar Melendo netted the victor for the hosts in the 88th with his first goal as a professional. I think it's a good step forward in conviction.
Real Madrid were without the rested Cristiano Ronaldo, Gareth Bale and Marcelo, with Karim Benzema also missing because of a hamstring injury. It had won 23 of its last 29 matches in all competitions.
Costa broke the deadlock with a low shot into the far corner after a picking up a loose ball inside the area. It should give us confidence to build something important.
Melendo netted the victor with a low shot from the middle of the area after a well-timed pass by Marc Navarro.
Atletico's lead did only last seven minutes, though, as Sevilla levelled things up 10 minutes from time when former Manchester City winger Jesus Navas saw his cross from the right flank take a wicked deflection which Moya could not stop from looping into his own net.
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As the minutes passed without a Real Madrid goal, Leganes dared to dream that they would finally reach the second leg of a Copa tie with something at stake. "They played in a very compact style, they're a daring team and we struggled to beat them".
The result ended Atletico's four-game winning run.
An Atletico fan was hospitalised following a violent incident before Wednesday's match.
Real struggled against a weakened Leganes team showing eight changes from the 3-2 La Liga defeat by Real Betis on Monday, indicating the host' lack of enthusiasm for the Cup despite supporters packing out their Butarque stadium.
Alaves played a man down from the 77th after defender Adrian Dieguez was sent off with a second yellow card.