Emiliano Armenteros' goal after 15 minutes set Javi Gracia's side on their way to victory in the coach's first home game in charge, and Oriol Riera extended the lead just a few minutes into the second half.
Edu Albacar scored a penalty for the visitors with three minutes of normal time remaining to set up a tense finish, but the Navarrans held on to pick up their first points of the season and end a four-game losing streak.
Gracia's side went into the game after losing at Getafe a week earlier, despite taking an early lead.
They were dealt a blow during the week when injury ruled midfielder Alvaro Cejudo out of the game, meaning Sisinio Gonazalez 'Sisi' started instead.
Osasuna were more physical than in their earlier games and showed signs of improvement in defence, but were also grateful to their goalkeeper Andres Fernandez, who pulled off three vital saves in the first half to preserve his side's lead.
Armenteros and Raoul Loe had Osasuna's first attempts of the game, neither of which found the target, but with a third attempt, the hosts took the lead.
Riera contested a high ball on the edge of the area and held the ball up for Roberto Torres, who battled with an Elche defender, the ball eventually falling to Armenteros on the left hand side of the box, and the Argentinian arrowed the ball in off the post first time.
Moments later Torres then had a chance to add a second goal, but was denied by Elche stopper Manu Herrera.
At the other end, Fernandez produced his first excellent save, tipping away a venomous shot from Carlos Sanchez on the edge of the area.
Four minutes later the goalkeeper made an even better stop, turning wide an unexpected early attempt from Richmond Boakye which was headed for the top corner.
And before half-time he had to make yet another save, this time denying Albacar.
Barely three minutes into the second period, the hosts doubled their lead.
Jordan Loties slipped the ball towards Riera in the box, the striker controlled with his left foot, shot against a defender and then volleyed the rebound with his right foot into the bottom corner.
Fran Escriba's side were knocked flat by the second goal, and struggled to break down an Osasuna side that was determined not to cede any ground.
Towards the end of the match substitute Javi Marquez made an audacious attempt to lob Fernandez from just ahead of the halfway line, but the shot crept just over the bar.
Elche eventually did get back into the game via the penalty spot. Marc Bertran blocked a cross with his hands, and Albacar sent Fernandez the wrong way to score with two minutes left of normal time.
The goalkeeper went down injured after the spot-kick, leading to four minutes of added time being played.
Elche were then awarded a corner in the final minute of injury time, but the Osasuna defence held firm.
Sevilla are now bottom of the league, but if results go against them, it could be Elche that is propping up the table come the end of the weekend.