Argyle captain Paul Wotton gave the Pilgrims the lead just before the interval after striker Barry Hayles had been pulled down by Wednesday keeper Brad Jones.
However, the Owls were not behind for long as substitute Sean McAllister - on for striker Wade Small - levelled with a well-executed strike early in the second half.
Despite coming under intense pressure from the home side, a Wednesday defence marshalled by former Pilgrim Graham Coughlan withstood the assault and broke away to net a deserved winner through South African midfielder Burton O'Brien.
O'Brien took the ball superbly on his chest and stabbed it past Argyle goalkeeper Luke McCormick after being played in well by midfielder Kenny Lunt.
Wednesday still had to endure a concerted effort from the home side - and more than seven minutes of injury-time - before confirming their first victory of the season.
Substitute Sylvan Ebanks-Blake, formerly of Manchester United had three chances to rescue a point for the Pilgrims.
First, he was denied by a fantastic save by Jones at the foot of the post, following Akos Buzsaky's right-wing free-kick and he followed that with a shot that went narrowly wide.
Then, after fellow striker Hayles had lobbed an effort on to the top of the bar, Ebanks-Blake wasted a glorious chance when he headed over from an excellent position in the centre of the goal.
Wednesday, who could have taken the lead early on when Small fired over, also had moments when their counter-attacks nearly gained reward.
Chris Brunt, playing in a central striking role after Small was withdrawn due to injury, hit the bar with an attempted lob and also saw a shot on the turn deflected just wide.