Spain striker David Villa set them on their way with his ninth of the campaign, acrobatically prodding home Carlos Marchena's lob over the defence in the 12th minute.
David Albelda and then Marchena both scored from outside the area with superb shots, and Osasuna, although they pulled one back through Masoud midway through the second half, finished in disarray with nine men on the pitch.
Unai Emery's well-organised side climbed up to fourth with 24 points from 11 games, pushing Deportivo La Coruna down to fifth on 22. Depor overcame struggling Atletico Madrid 2-1 on Saturday.
The in-form Nilmar, who scored in Brazil's recent friendlies against England and Oman, headed the opener after seven minutes and looped in a second header just after the re-start.
Italy's Giuseppe Rossi made it 3-0 before Diego Costa pulled one back for the visitors, but Ernesto Valverde's side continued their steady climb up the table after being rooted to the bottom last month.
