The Dutch forward, on as a substitute, prodded home in the final stages to help Atletico join Real Sociedad, Athletic Bilbao and Real Mallorca in the final four.
Antoine Griezmann skied a first-half penalty for Diego Simeone's side who ground out their win against their resilient visitors at the Metropolitano stadium.
Atletico survived a late scare when Pablo Barrios conceded a penalty but the decision was overturned after the referee reviewed the incident with VAR.
Players observed a moment of silence before the game to pay respect to three Sevilla fans who died in the morning in a traffic accident on the way to Madrid.
Sevilla defender Sergio Ramos, formerly captain of Atletico's rivals Real Madrid, came in for jeers from the start and almost silenced them as he narrowly failed to reach a free kick.
Quique Sanchez Flores' Sevilla, struggling badly in La Liga, made a strong start but gave away a penalty when Marcao clumsily fouled Nahuel Molina.
However, Griezmann slipped as he shot, sending the ball wildly looping off target.
Sevilla sat deep to defend and frustrate the hosts, who failed to create any further clear chances until late on.
Griezmann had a goal ruled out for offside before Simeone took him and strike partner Alvaro Morata off after 66 minutes to shake up his attack, with Depay and Angel Correa coming on.
Depay quickly struck and although it was chalked off for offside, eventually he did break the deadlock.
Correa cut in well from the right and slid the ball into the area for Depay to poke home after 79 minutes as Simeone's substitutes teamed up.
Ramos had the chance to equalise late on but could only send the ball over from point-blank range as he lunged in desperately.
Atletico midfielder Barrios lunged in desperately too, on the edge of his own box deep in stoppage time, appearing to trip Erik Lamela, with the referee pointing to the penalty spot.
The official, Jesus Gil Manzano, was called by VAR to review the incident and changed his mind, letting Barrios and Atletico off the hook.
Gil Manzano apparently decided Barrios touched the ball, although it was not immediately clear if he did from replays, and Sevilla players remonstrated with the referee at full time.
