The home side dominated the first half but failed to score, the best chance coming from a penalty for a foul that Antoine Griezmann failed to convert after slipping.
The French forward had a chance to redeem himself in the second half, but his effort was disallowed for offside in the 60th minute and Atletico had another effort ruled out after a Depay breakaway.
Depay found the net in the 79th when Angel Correa's through ball slipped between Sergio Ramos's legs to secure the win.
Sevilla could have equalised in stoppage time after Pablo Barrios's slide tackle on Erik Lamela earned them a penalty, but a VAR review overturned the decision.
"That last play showed Pablo (Barrios) not to take unnecessary risks. It's good that it happened to him and that there were no consequences in the end," said coach Diego Simeone.
"We have made a great effort to be where we are and in the second half we had a different tempo, we played more risky and the changes gave us the jump we needed," he added.
They are joined by Athletic Bilbao, who reached the semi-finals by  knocking out Barcelona 4-2, Mallorca and Real Sociedad.
Mallorca eliminated LaLiga leaders Girona 3-2 while Real Sociedad triumphed 2-1 over Celta Vigo.
The semi-final draw will be made on Friday.
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