On loan from Barcelona, Abde skillfully broke through Athletic's defence and struck the ball home, giving his team a slight advantage ahead of the second leg on April 4 at San Mames.
Osasuna, who haven't reached the semifinals since losing to Real Betis 18 years ago, will be hoping to advance to the final with this slender lead.
Athletic Bilbao, on the other hand, have an impressive record of 23 cup wins, second only to Barcelona, but their last victory was in 1984, and they have lost in six finals since then.
Despite a few early chances, the first half was relatively uneventful.
However, Abde's moment of brilliance proved to be the game's turning point.
Osasuna coach Jagoba Arrasate acknowledged that both teams played well but Abde's goal gave them the upper hand.
In the final stages of the match, Athletic pushed forward in search of an equalizer but Osasuna's defence held firm.
Sergio Herrera made a crucial save to deny Athletic striker Gorka Guruzeta, keeping his team's clean sheet intact.
Overall, the match was evenly contested, and both coaches agreed that there weren't many clear-cut opportunities.
Ernesto Valverde, Athletic's coach, lamented that the game was "scrappy," but he also acknowledged Osasuna's impressive performance.
In the other semifinal first leg on Thursday, Real Madrid will host their fierce rivals Barcelona.