Adam Armstrong sent the Saints into the lead in the eighth minute of the contest, but it did appear that the spoils were going to be shared, with Lee Gregory levelling the scores in the 54th minute.
Adams had other ideas, though, coming up with Southampton's second in the 87th minute to ensure that the relegated outfit made the perfect start to life back in the second tier of English football.
James Ward-Prowse captained the Saints amid the speculation surrounding his future, and it was the captain that came up with the assist for his side's second in the latter stages during a brilliant performance.
Initially it appeared that Nathan Tella had sent Southampton ahead, with his curling effort from outside the box finding the back of the net, but it was ultimately decided that it had taken a flick off Armstrong, with the striker opening his tally for the season in bizarre circumstances.
Russell Martin's side threatened to double their advantage through Ward-Prowse and Ryan Manning, while Carlos Alcaraz also came close with a strike in the 22nd minute.
Wednesday's best chance of the first half fell to Tyreeq Bakinson in the 33rd minute, but his effort from a Barry Bannan pass missed the target, and the home side were deservedly behind at the break against their slick opponents, who will fancy their chances of making a strong push for promotion this term.
The Owls managed to level the scored in the 53rd minute, though, with Gregory firing home from inside the box after Southampton had only half-cleared a corner.
Samuel Edozie and Tella came close just past the hour, with Southampton looking the more likely, and the strings continued to be pulled by Ward-Prowse, who starred for the visitors.
Wednesday looked to be holding on for a welcome point in their Championship opener, but Southampton secured the win in the 87th minute through Adams, who turned a low Ward-Prowse cross into the back of the net from close range to silence the home side's supporters.
There were nine minutes of additional time at Hillsborough, but Wednesday never looked like scoring an equaliser, with Southampton, who had Romeo Lavia as an unused substitute amid the speculation surrounding his future, making the perfect start to their campaign.
Next up the Saints is a trip to Gillingham in the first round of the EFL Cup on Tuesday, while Wednesday will host Stockport County in the same competition on the same night.