Aritz Aduriz scored twice, taking the match away from the hosts with his side's second and third goals after Markel Susaeta's opener had been cancelled out by Emiliano Armenteros.
Osasuna were dealt another setback by the 76th-minute sending-off of Alejandro Arribas and Athletic took full advantage.
Substitutes Ibai and Enrique Sola struck late on as the visitors strengthened their hold on fourth place in the Primera Division.
Athletic got off to a flying start and went ahead after only three minutes, Susaeta firing a fierce shot into the top corner.
Osasuna levelled in equally emphatic fashion in the ninth minute as Armenteros found the net with an expertly-taken free-kick.
The hosts were back in front seven minutes later when Aduriz pounced to knock home the rebound after goalkeeper Andres Fernandez had done superbly to parry Ander Herrera's close-range header.
Both sides continued to pose a threat, but it was Bilbao who looked to have the greater cutting edge up front.
Fernandez did well to push away an Aduriz shot from the right side of the area in the opening moments of the second half.
But the striker got his second just after the hour mark, heading a Herrera cross into the bottom corner.
Athletic were well on top now and their opponents' task got even harder when they were reduced to 10 men, Arribas picking up a second yellow card for a foul on Susaeta.
Ibai added to the agony with a fourth in the 84th minute, Aduriz turning provider and laying on the goal for his team-mate, who fired into the roof of the net.
Sola made it five with a close-range finish two minutes from time and Iker Muniain almost grabbed another as Athletic finished with a flourish.