Marcelo Bielsa's side will be confident of overcoming the Islanders in the return leg as they extended their six-year unbeaten home record in the Copa del Rey
Fernando Llorente's powerful header midway through the first-half conquered a resolute Mallorca defence, and the Basque side's lead was doubled on the hour mark. In a replica of the first, Iker Muniain directed the ball beyond the grasp of Juan Calatayud as Marcelo Bielsa's side moved into an ominous position ahead of the second encounter.
Mallorca successfully negotiated the opening exchanges before Gorka Iraizoz thwarted the Islanders when they began to exert their attacking intentions. The Basque keeper readjusted quickly to palm Gonzalo Castro's deflected shot around the post.
The home side began to dominate possession as they fashioned numerous chances, the best of which fell to Jon Aurtenetxe. The full-back was however denied by a reactionary save from Calatayud as the Islanders defence remained strong.
In an enthralling period possession switched sides in quick succession before Michael Pereira evaded the Athletic defence. He bore down on Iraizoz's goal but the keeper did well to divert the Frenchman's shot across goal away to safety.
Athletic Bilbao responded emphatically through Llorente's clinical header after Oscar de Marcos had found enough room to dink a cross onto his strike partners head, giving the Basque a slender lead at the break.
Inigo Perez was introduced at half-time and was soon testing the palms of Calatayud when the Israeli stopper tipped the substitutes' free-kick around the post.
The Basque doubled their lead on the hour mark when Muniain was left unmarked at the far post, forcing the Islanders to abandon their defensive instincts.
Mallorca searched for a away goal in the closing stages at San Mames but were unable to penetrate a solid Bilbao defence with Iraizoz untroubled.
Conversely, it was the Northerners who came closest to extending their lead in the final minutes and Bielsa's side were left rueing a disastrous decision from the officials who ruled out a perfectly legitimate goal for offside.
Bielsa will be confident of leading his side into the semi-finals of the Copa del Rey with either Espanyol or third-tier Mirandes awaiting his side.
The Islanders will hope to reverse the first-leg scoreline when they meet in seven days time at the Ibeorstar Stadium for the return fixture.