Argentine striker Ezequiel 'Chimy' Avila gave Osasuna a ninth-minute lead.
However, the advantage lasted just two minutes before Rafa Mir made the most of a fine pass from another Argentine, Papu Gomez to equalise.
Sevilla finished last season in fourth spot in LaLiga, 27 points ahead of Osasuna in 10th.
But the underdogs were not overawed and grabbed a 74th-minute winner from the penalty spot courtesy of Aimar Oroz.
Sevilla lost centre-backs Jules Kounde and Diego Carlos to Barcelona and Aston Villa respectively in the summer.
On Friday, they were also without veteran striker Isco who arrived on a free transfer from Real Madrid but wasn't considered ready to make the match-day squad.