The two teams kept their fiery excitement until the 84th minute, when Juanmi led Betis in the score with the first goal after getting into 10 minutes.
But Athletic Bilbao equalized in the last breath by veteran Raul Garcia, who equalized in the fourth minute of stoppage time, saving the Basque Lions from a detective elimination.
In the penalty shootout, Garcia scored again, in addition to Inaki Williams, John Murcio and Yuri Perchichi, while Betis scored only Issa Mandi, and Sergio Canales and Juanme were wasted.
To reach Athletic Bilbao to the semi-finals for the second consecutive season, after qualifying to the final of last season, which is scheduled to be played in about two months.
Athletic Bilbao joins the pre-qualified trio: Levante, Barcelona and Seville.
The semi-final draw will be held on Friday at the Spanish Football Federation headquarters in Las Rosas.