Inaki and his younger brother Nico were both on the score sheet as Ernesto Valverde’s side coasted to a 3-0 win at home.
Athletic Bilbao had beaten Diego Simeone’s side 1-0 in the first leg and completed the double over their opponents in the second leg on Thursday.
Inaki opened the scoring in the 13th minute when he applied a beautiful volley to Nico’s pin-point cross from the left flank.
The Williams brothers again combined to double Athletic Bilbao’s lead, this time Inaki setting up Nico to make it 2-0.
Gorka Guruzeta added a third after the hour mark to complete the rout as Valverde’s side advanced to the final of the Copa del Rey, where they’ll face Real Mallorca.
Meanwhile, Inaki has now scored 11 goals and provided five assists in 27 matches in all competitions this season.
The 29-year-old has been in imperious form since returning from the 2023 Africa Cup of Nations (AFCON) in Ivory Coast, where Ghana exited at the group stage.
His brother Nico, who opted to play for Spain, has also been involved in 17 goal contributions this season (6 goals and 11 assists).
Athletic Bilbao will have the chance to win a trophy when they face Mallorca in the Copa del Rey final in April.