The hosts claimed the win at Home Park despite being reduced to 10 men in first-half stoppage time.
The Pilgrims went ahead in the 19th minute of the League One clash after Wednesday midfielder Darren Potter slipped while making a pass in the centre of the field. The ball ran to Bradley Wright-Phillips who raced forward down the left and beat two players before teeing up skipper Carl Fletcher on the edge of the box.
The visiting Owls levelled in the 38th minute through midfielder James O'Connor, who powered home a far-post header from Tommy Spurr's deep cross.
Argyle saw right-back Bondz N'Gala dismissed for a foul on Wednesday striker Neil Mellor in a one-on-one situation in first-half injury time.
Wright-Phillips then put Argyle back in front in the 46th minute, heading home a left-wing cross from Noone.
Wednesday substitute Miller levelled after 75 minutes with a cracking long-range half-volley. But their joy was shortlived as two minutes later Jim Paterson - one of five players recalled for Argyle - beat O'Connor down the left wing and crossed to the far post where Noone headed home.