Espanyol started the game brightly and could have been ahead when Joan Verdu found space in the corner of the box, but his effort was well saved by Andres Fernandez.
The visitors took the lead midway through the first half as Alvaro Cejudo latched on to a loose ball in the area to fire past Cristian Alvarez.
Osasuna looked comfortable following the goal and were still dominant in possession as the second half began.
However, Verdu once again threatened for the hosts by hitting the post with a powerful strike from just outside the box.
Espanyol were punished for not taking the chance almost immediately when Kike Sola doubled the lead from close range following good work from Cejudo.
Manuel Omwu made sure of the result with 10 minutes remaining after coming off the bench to drill the ball into the bottom corner from 20 yards out.