Marcus Tudgay was the main architect of the Pilgrims' demise as his header set up Richard Wood to give Wednesday a first-half lead before scoring from the spot in the second half.
Alan Gow's excellent free-kick gave Plymouth hope late on but Tudgay wrapped up the win in stoppage time.
Plymouth started well and had a good chance to go ahead after only four minutes when James Paterson's corner found Cillian Sheridan but the on-loan Celtic striker headed over from six yards.
The visitors were struggling to get going and their cause was not helped as Michael Gray was forced off injured and replaced by Sean McAllister.
Wednesday did eventually start to pose an attacking threat but the Plymouth defence was restricting them to long shots and James O'Connor sent a volley wide from distance.
Minutes later the visitors took the lead with their first meaningful chance of the game as Darren Potter's corner found Tudgay, who cleverly headed across goal and Wood was on hand to sweep the ball home.
The Owls immediately set about doubling their lead with Jermaine Johnson trying his luck from 25 yards but Plymouth goalkeeper Romain Larrieu was equal to the effort.
With half-time approaching Tudgay found space inside the box but drilled his shot over the bar when he perhaps should have tested Larrieu.
At the other end O'Connor brought down Jamie Mackie 30 yards from goal and Alan Judge sent the resulting free-kick just wide of the post.
Plymouth began the second half with ominous intent and were desperately unlucky not to draw level in the 54th minute when Carl Fletcher drilled a fine effort from 25 yards which cannoned agonisingly off the post.
Minutes later Gow fired wide from a free-kick before Sheridan wasted another chance when he dragged his shot wide from 12 yards.
However, just as in the first half the visitors made Plymouth pay for their profligacy when Marcel Seip hauled down Luke Varney in the area and Tudgay netted his second league goal in as many games from the spot.
With 10 minutes remaining the Pilgrims were given a lifeline when Tommy Spurr handled and Gow curled in a superb free-kick from 25 yards.
Paul Sturrock's side pushed forward in search of an equaliser but with the full-time whistle approaching Tudgay latched onto Potter's pass and produced a tidy finish past Larrieu.