The five-time winners of the competition led 2-1 heading into Tuesday's second leg after inflicting a first domestic defeat on their opponents at the Wanda Metropolitano last week.
An early goal from Sergio Escudero in front of a packed Ramon Sanchez Pizjuan crowd put Los Rojiblancos on their way, with Ever Banega and Pablo Sarabia adding to the scoring after Antoine Griezmann had temporarily levelled things up on the night.
Sarabia lofted in a cross for Escudero to meet on the volley inside the opening minute in Andalusia, leaving stand-in keeper Miguel Angel Moya with little chance of keeping it out.
With a two-goal deficit to overcome, Atletico - nine points better off than their opponents in the La Liga standings - plugged away and had a route back into the game through a stunning Griezmann strike.
Sevilla had not lost to Atleti in eight previous Copa home ties, though, and it was the home side who scored the pivotal third goal of the evening early in the second half.
After Gameiro and Griezmann squandered good chances to convert, both at the end of Sime Vrsaljko crosses, Saul Niguez barged into Joaquin Correa to gift Banega a chance to convert from the penalty spot.
Visiting boss Diego Simeone rolled the dice with his changes, but the gamble did not pay off as Sarabia was picked out on the counter and placed the ball out of Moya's reach after cutting onto his left foot.
Yannick Ferreira Carrasco was next to be denied by Rico from the final opening of the match, meaning that Sevilla advanced through with some room to breathe.
SEVILLA (4-2-3-1): Sergio Rico; Navas, Mercado, Lenglet, Escudero; Banega, N'Zonzi; Sarabia (Ben Yedder 73), Vazquez (Pizarro 84'), Correa (Nolito 87'); Muriel
ATLETICO MADRID (4-4-2): Moya; Vrsaljko, Godin, Gimenez (Thomas 64'), Lucas; Correa (Carrasco 60'), Saul, Gabi (Torres 56'), Koke; Gameiro, Griezmann