Celta's Iago Aspas was denied by the woodwork in the 67th minute with a bullet strike from the heart of the area on a rain-soaked pitch at the Balaidos stadium.
Alaves, the 2001 UEFA Cup finalists, who are attempting to reach the King's Cup final for the first time, will host the second leg on Feb. 8.
Three-times finalists Celta, who knocked out Real Madrid in the quarter-finals, dominated possession for most of the first half only for Alaves to create the first chance in the 40th minute.
The visitors launched a counter-attack that ended with Manu Garcia getting on the end of Victor Camarasa's cross and striking a left-footed effort towards the near post that drew a flying save from Celta goalkeeper Sergio Alvarez.
Celta almost broke the deadlock on 58 minutes but Alaves goalkeeper Fernando Pacheco kept out Aspas's shot from six metres.
Alaves substitute Edgar Mendes had a one-on-one with Alvarez but shot straight at the keeper in the 78th minute.
Barcelona won 2-1 at Atletico Madrid on Wednesday in the other semi-final, ahead of next week's return leg at the Nou Camp.
(Reporting by Adriana Garcia; Editing by Toby Davis)
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