Fernando Llorente scored twice in four second-half minutes, after Mirandes had pulled one back through defender Aitor Blanco, who also grabbed their second near the end.
Cesar Caneda put the ball in his own net two minutes from time as Bilbao, who have won the Cup 23 times, progressed 8-3 on aggregate.
Marcelo Bielsa's side will find out their final opponents on Wednesday, when Valencia visit the Nou Camp to play Barca with the tie level at 1-1.
"It's incredible to be in another final after only a few years, we are delighted," Llorente told Spanish television.
"We knew we couldn't afford to relax at all, and the few times we did, they scored.
"I don't care who we meet in the final as long as we win. I had a bad experience with Barca, but we'll give it our best whoever it is."
Bilbao lost 4-1 to Barca in the 2009 Copa final.
Mirandes, who play in Spain's Segunda B, knocked out top-flight sides Villarreal, Racing Santander and Espanyol to reach the last four, and were the second team only from the third tier to do so in the competition's current format.
Bilbao never gave the visitors time to settle, however, and a failure to clear a cross in the 11th minute allowed Muniain to burst through the middle and fire in the opener.
Mirandes were still reeling three minutes later when the hosts strung together a series of passes that pulled the defence apart and Susaeta turned to slot away the second.
Aurtenetxe looped in a header to make it 3-0 and Mirandes did not manage to test Bilbao keeper Gorka Iraizoz until the 35th minute when Pablo Infante drilled a shot straight at him.
Mirandes showed some of the courage and talent that have made them headline news in Spain after the break, and were rewarded in the 58th when a rebound from a shot against the post was turned in by Blanco.
They flashed another effort wide of the post before in-form Spain striker Llorente lobbed the visiting keeper from outside the area in the 71st. He then slid in at the far post to notch his eighth goal from the last four matches.
Both sets of fans cheered their teams in the closing stages and Mirandes even hit the post before the referee blew the final whistle.
"It's been an unforgettable experience. Something we'll remember for the rest of our lives," Mirandes captain Pablo Infante said. "Now we need to focus on winning our league and getting promotion."
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