Ernesto Valverde's side brought a 1-0 lead into the second leg of the contest at San Mames, with an Alex Berenguer penalty proving to be the difference between the two sides in the Spanish capital.
The Lions were able to finish the job in front of their own supporters on Thursday evening, with Inaki Williams, Nico Williams and Gorka Guruzeta on the scoresheet in a 3-0 success for the Basque outfit in the second leg.
Athletic, who are the second-most successful side in the history of the Copa del Rey with 23 titles, will take on Mallorca for the trophy at Estadio de La Cartuja on April 6, with Javier Aguirre's side dumping Real Sociedad out of the competition on penalties on Wednesday night.
Atletico were unable to call upon Antoine Griezmann due to an ankle problem which he picked up in last week's Champions League last-16 first leg with Inter Milan, and the Frenchman's invention and work-rate was missed badly by Diego Simeone's side on a difficult night.
The visitors actually had the first opportunity of the match in the sixth minute through Samuel Lino, who had broken into a dangerous position inside the Athletic box, but his effort was kept out by Julen Agirrezabala.
Athletic made the breakthrough on the night in the 13th minute, though, with Nico Williams setting up his older brother Inaki Williams, who volleyed one past Jan Oblak to spark jubilant scenes inside the stadium.
Angel Correa, who benefited from Griezmann's absence to start, looked to hit back for Atletico in the 17th minute, but his effort was always rising over the crossbar.
Correa had another chance in the 31st minute after an excellent pass from Alvaro Morata, but the Argentine again fired over the crossbar in what was a let-off for the home side.
Athletic doubled their lead on the night and made it 3-0 on aggregate in the 42nd minute, with Inaki Williams this time turning provider for Nico Williams, who found the back of the net from a cut-back into the box.
Atletico almost had one back just before the interval, with Stefan Savic rising highest inside the penalty box to meet a corner, but the centre-back's header just missed the post.
Nico Williams and Oihan Sancet both had chances to add to the scoring early in the second period, but the former was denied by Oblak, before the latter missed the target from a promising position.
Athletic made sure of their progression to the final just past the hour, though, with Guruzeta firing into the back of the net from close range after Oblak had kept out a strike from Sancet.
Saul Niguez almost scored a consolation for Atletico late on, but he headed wide from close range, and it finished 3-0 on the night, with Valverde's side comfortably progressing to the final courtesy of a 4-0 aggregate success.
Both teams will switch their attention back to La Liga this weekend, with fourth-placed Atletico facing a home game against high-flying Real Betis, while Athletic, who are only three points behind Atletico in fifth, have the small matter of a clash with the champions Barcelona at San Mames.