Igor Gabilondo and David Lopez gave Bilbao a 2-1 aggregate win away to Sporting Gijon, that earned them a meeting with either Sevilla or holders Valencia in their semi-final, first leg next week.
Valencia visit Sevilla on Thursday holding a 3-2 first-leg advantage.
The 2003 Cup winners Mallorca held Real Betis 0-0 in Seville to go through 1-0 on aggregate, and will face either Barcelona or Espanyol in the last four.
Primera Liga leaders Barca play Espanyol at the Nou Camp on Thursday after a goalless first leg.
Bilbao, who won the last of their 23 King's Cups in 1984, were behind from inside the first minute in Gijon, when Sporting's Carmelo fired in from the edge of the area.
Bilbao came close on a number of occasions in an open encounter but it was Gabilondo who levelled before the break, lobbing Sergio Sanchez after chasing a flick-on.
Sporting's defence, that was breached five times away to Getafe in the league on Sunday, went missing six minutes after the re-start gifting Lopez a free header that settled the tie.
Mallorca are in the relegation places in the Primera Liga but they were good for their 0-0 draw away from home, as Alhassane Keita's first-leg goal proved enough to put them through.
However, the Cup exit increases the pressure on Betis coach Paco Chaparro, whose side have lost their way in recent weeks and now sit four points above the bottom three in the league.
