Roberto Fernández struck twice inside the opening 17 minutes at El Sadar, first converting from close range after a rapid four-pass move down the right, then pouncing when Aitor Catena failed to clear a low cross into the box.
Osasuna also had a strong claim for a penalty waved away by referee Luis Bestard, and not reviewed by VAR, after Riedel appeared to trip Raúl García de Haro inside the area, while a Rubén García goal was later ruled out for an offside throw-in.
Manolo González was satisfied with a complete performance from his side, even if he stopped short of calling it perfect.
"It has been a good match. I don't know if it was a complete performance or not, because we always tend to focus on the negative," González said. "It's been a complete game, on a difficult pitch, against an opponent that demands a lot of you in terms of crosses and presence in the box.
"We're on matchday five and there's still a long way to go and a lot of work to do."
The Espanyol coach credited a week of specific tactical preparation for both goals.
"The goals were situations we had worked on, but on the right side rather than the left," he said. "One of the goals came from a ball to Edu, who found space and played the pass into the area for Omar. Luckily it worked out well, and that's what we're looking for: for the team to improve day by day in situations of attacking the box and finding space behind the opposition defence."
Osasuna head coach Luis Miguel Ramis was scathing in his self-criticism over the defensive errors that led to both goals.
"We tried, of course. Maximum self-criticism for errors that shouldn't happen, because once again they scored two goals against us too easily," Ramis said. "The match is defined by the penalty decision. The VAR's decision is absolutely inexplicable."
Ramis said he had believed a single goal could have shifted the momentum back in his side's favour.
"I had it in my head that with one goal the team would react, and we thought it was possible to turn the match around, but it never came," he said. "Our fans are truly incredible, they give us so much strength. They support us throughout the whole match and afterwards too. I've never seen anything like it in my life and we cannot let them down. We are not going to."