Their cause was helped by the first-half dismissal of Francis Jeffers for what appeared to be a head-butt on Vale midfielder Tommy Fraser, and two goals in three second-half minutes from Kris Taylor and Robert Taylor earned them victory.
Vale started brightly, and Marc Richards should have done better after good link-up play involving Gareth Owen and Kris Taylor but he screwed his shot wide.
The visitors' response came from Tommy Miller but the midfielder's 20-yard drive was comfortably gathered by Chris Martin.
Valiants centre-back John McCombe sent a header over the crossbar from six yards after 20 minutes, while Robert Taylor drilled a shot over the crossbar from just inside the area on the half-hour as the home side dominated.
Louis Dodds forced a fine save from Lee Grant in the 34th minute, and Vale's chances of pulling off another shock increased shortly afterwards when Jeffers was sent off.
The Owls striker was involved in an off-the-ball incident with Fraser and left the field with blood pouring from a head wound.
Brian Laws brought on Jermaine Johnson and Akpo Sodje at the break, and the latter was immediately involved, forcing a fine save from Martin with a close-range header.
Wednesday forward Marcus Tudgay had Martin scrambling to keep out a well-struck shot from 20 yards, while at the other end Richards brought a decent stop from Grant.
Vale went ahead after 62 minutes when a corner on the right was taken short to Kris Taylor and he worked some space before unleashing a fierce shot into the top corner.
Robert Taylor stunned the visitors as he extended the Valiants' lead three minutes later, hooking home a superb cross from Dodds.
Sheffield Wednesday rarely threatened in the closing stages, with a weak shot from Darren Potter the closest they came to a consolation.