After last week's first leg in Pamplona finished 1-1, a fired-up Athletic carved out a 3-1 aggregate lead inside the first 20 minutes at the San Mames stadium.
Gabilondo opened the scoring on five minutes, controlling an attempted clearance and curling the ball over Ricardo Lopez, before a low shot from striker Ion Velez skewed off Josetxo and over the stranded Osasuna goalkeeper into the net.
The visitors had a chance to get back into the match when Gabilondo fouled Javier Portillo in the area with around half an hour left but the striker had his penalty well saved by Athletic keeper Gorka Iraizoz.
Osasuna finished the match with 10 men after French midfielder Ludovic Delporte was dismissed for a wild challenge on Markel Susaeta in the 90th minute.
The defeat capped a miserable few days for the struggling club, who surrendered a 2-1 lead to Barcelona in the final 10 minutes of Sunday's Primera Liga match to lose 3-2 and remain rooted to the bottom of the standings.
Athletic will play Sporting Gijon in the last eight after Jorge Garcia's header in added time on Wednesday sent them through at the expense of Real Valladolid.
In the other three quarters, also decided on Wednesday, Espanyol face city rivals Barcelona, Real Mallorca take on Real Betis and holders Valencia play 2007 winners Sevilla. The two legs are provisionally scheduled for January 21 and January 28.
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