One-off matches have replaced two-legged ties until the semi-final stage this year, with the games played at the home of the lower-division teams.
That saw both Barcelona and Real Madrid survive some nervy moments before advancing on Wednesday against Ibiza and Unionistas de Salamanca, respectively, but Atletico were not so fortunate.
Diego Simeone's side went ahead through Angel Correa after 62 minutes and looked set for a hard-fought victory until Julen Castaneda levelled late in the game.
Extra-time followed and there were chances at both ends, but the Leon-based side came out on top as Sergio Benito beat Antonio Adan after 108 minutes.
Cultural Leonesa survived a late onslaught from the Rojiblancos and clung on for a famous win that sees them advance to the last 16.
In a night of shocks, Segunda side Rayo Vallecano knocked out Real Betis on penalties following an exciting 2-2 draw in which Andres Martin had forced the shootout with a leveller two minutes before the end.
Mirandes (also in Segunda) beat Celta Vigo 2-1 and Segunda B outfit Badajoz overcame Eibar in a 3-1 win.
The draw for the last 16 takes place at midday GMT on Friday.