Diego Simeone's side looked on course to take a first-leg lead into the halfway stage of the tie when Diego Costa continued his good start to life back at the club with the opening goal just 17 minutes from time.
The hosts thought that they had taken the lead after only 13 minutes when Costa planted his header past Sergio Rico in the Sevilla goal, only for the referee to controversially rule it out for a foul.
Costa was not deterred, though, and the Spain international came close again just six minutes later with another header from close range, although this time Rico got across to make a fine save.
It took until four minutes before half time for Sevilla to have their first clear opening of the contest when Sergio Escudero let fly with an effort which looked destined for the top corner until Moya pulled off a flying save to turn the shot around the post.
Costa continued to look the most likely to break the deadlock, though, and he did so in unerring fashion in the 73rd minute when the ball broke to him inside the box and he rifled a low left-footed strike across goal and into the bottom corner.
Sevilla almost responded immediately when Escudero tried his luck from range again, only to see his powerful strike fly narrowly past the target just a minute after his side had fallen behind.
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.
It was a hugely unfortunate goal for the home side to concede having kept six clean sheets in their last seven outings, but another arrived just eight minutes later when Correa pounced on a loose ball inside the area and slotted his finish past the keeper.
The visitors then held out through four minutes of stoppage time to end Atletico's unbeaten home run in domestic matches, with their only previous defeat in their new home having come at the hands of Chelsea in the Champions League.
The two sides will face off again in a week's time, when Atletico will need to come from behind if they are to reach the semi-finals for a second consecutive season.
ATLETICO MADRID (4-4-2): Moya; Juanfran, Savic, Godin, Hernandez; Koke (Torres, 82'), Gabi, Niguez, Vitolo (Correa, 46'); Costa, Griezmann (Carrasco, 68')
SEVILLA (4-2-3-1): Rico; Corchia (Geis, 90'), Mercado, Lenglet, Escudero; N'Zonzi, Banega, Sarabia, Vazquez (Navas, 75'), Correa; Muriel (Ben Yedder, 84')