Jagoba Arrasate's men moved to the brink of promotion with a 0-0 draw away to Cadiz.
When Albacete went down 1-0 against Granada a day later, Osasuna's top-two finish was secured.
"We are very happy, I think we are here on our own merits," captain Roberto Torres told El Larguero. "We will give more value [to the achievement] in the days [to come]. The season has been beautiful and I think it is a beautiful moment at this time.
"We knew what could happen [when Albacete played Granada], we were aware that it could be a beautiful day."
Granada is highly likely to follow Osasuna into LaLiga as it is six points clear of third-placed Albacete, with three games remaining in the Segunda Division season.
Osasuna fans flood the streets of Pamplona
Arrasate's team could secure the title this weekend when they entertain Las Palmas, holding a four-point lead over Granada.
Osasuna was relegated after coming 19th in LaLiga in 2016-2017 and could only manage eighth in the second tier last time around.