It was the away side who started the stronger and Manuel Omwu fired wide after just four minutes when he might have scored.
Chances continued to come for both sides in the latter stages of the half, with Xavi Torres and Barkero leading the home side's charge, while Ivaro Cejudo almost added Osasuna's second minutes before the break.
Torres twice came close for the home side early in the second period, firstly with a header which was pushed away by Andres, before he fired his second effort past the post.
Levante continued to push for the leveller as the second half advanced, with Kone looking the most likely, but Osasuna stood strong to the constant pressure.
Nabil El Zhar might have levelled the match after 76 minutes, but he struck wide from inside the box.
Nino secured all three points for Osasuna, however, after 79 minutes to dent the home side's Champions League hopes.