A comfortable 2-0 victory for La Real sent them through to the quarter-finals while Atletico are left needing to beat Manchester United in the Champions League last 16 to have any chance of rescuing their season.
Diego Simeone's team are effectively out of the La Liga title race already given they sit 16 points behind Real Madrid and they were beaten by Athletic Bilbao last week in the semi-finals of the Spanish Super Cup.
The Copa del Rey appeared their only realistic chance of silverware left but instead they were convincingly beaten at the Reale Arena, where goals from Adnan Januzaj and Alexander Sorloth were enough to put the Basque side through.
Atletico Madrid have now managed only two victories in their last eight games in all competitions. In the league they have registered just one clean sheet in seven.