The Owls have now lost four games in a row after initially beginning life in League One by soaring to the top of the table, while the Saints had not won since sacking Alan Pardew despite being billed as two of the favourites for promotion.
Barnard's effort came about after some woeful Wednesday defending which started when Lewis Buxton put in a short pass to Darren Porter which the Saints exploited fully to give the visitors a lead they never looked like surrendering.
Rickie Lambert ran at the Wednesday defence before sliding across to his strike partner to clip the ball into the net beyond goalkeeper Nicky Weaver in the 61st minute.
Weaver was the busier of the two keepers throughout to deny Lambert and Barnard in particular, while substitute Adam Lallana twice threatened to add to Southampton's tally in the last 10 minutes.
Marcus Tudgay and Clinton Morrison had opportunities in the dying minutes to nick an equaliser for Wednesday but Kelvin Davis and his defence stood firm to spark wild celebrations at the full-time whistle.