Oriol Riera was next to come close for the home side, but his header whistled over the crossbar after 20 minutes.
Sevilla were ahead less than 10 minutes later when Jairo carried possession into the box before finding the bottom corner.
Fazio came close to making it 2-0 in the latter stages of the first period, but his header sailed over the crossbar after finding space inside a crowded box.
The visitors had their second in first-half stoppage time, however, when Bacca converted from the penalty spot after Damia had handled the ball inside the box.
Osasuna pushed for their opener in the second period and came close through Alvaro Cejudo and Armenteros, but it was Sevilla who passed up a golden chance just past the hour mark when Bacca fired wide from close range.
Bacca had another clear chance moments later after collecting a loose ball inside the box, but his effort rattled off the post as Osasuna struggled in defensive areas.
Osasuna registered late on through Acuna, but it was too little too late as Sevilla held on for all three points.