Mauricio Pellegrino's side will play 28-time Spanish Cup winners Barcelona in the May final after Luis Enrique's side beat Atletico Madrid 3-2 on aggregate on Tuesday.
Celta had conquered Real Madrid in the quarter-finals but never really got going at a packed Mendizorrotza Stadium that was rocking for Alaves's biggest game since their 5-4 defeat by Liverpool in the 2001 UEFA Cup final.
Spain international Iago Aspas had Celta's only real chances, seeing his low shot tipped away by the outstretched hand of goalkeeper Fernando Pacheco and watching an attempted lob land inches over the bar.
Alaves's Ibai Gomez clipped the bar at the other end in a cagey first half but the Basques dominated the second period and should have taken the lead when Brazilian forward Deyverson burst towards the area unchallenged but blasted over.
The breakthrough came with the aid of two headers, one from Deyverson and the other from Victor Camarasa which allowed Mendez to race into the area and lift the ball over Sergio Alvarez and into the net.
Alaves could have put the game beyond doubt but had to withstand six nervy minutes of stoppage time before their place in the final was confirmed.
Alaves beat Barca 2-1 at the Nou Camp earlier this season and will host Luis Enrique's side on Saturday in the league.
"The fans deserve this, they pushed us so hard against a really difficult team. It was a very equal tie and these fans pushed us over the line," Alaves captain Manu Garcia told Gol TV.
"We're going to enjoy tonight and we'll try to go for glory in May but we've already made history."
(Reporting by Richard Martin, editing by Ed Osmond)
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