The League Two side - who shocked Sheffield United with a 2-1 win at Bramall Lane in the first round - stunned their city rivals with two goals in the space of three minutes in the second half.
The Championship side had made life difficult for themselves as former Everton striker Francis Jeffers was sent off for an off the ball incident with Vale skipper Tommy Fraser.
Vale, managed by Sheffield-born Micky Adams and a self-confessed Blades fan, started brightly with defender John McCombe heading over the bar from a lofted cross from Kris Taylor on 20 minutes.
Brian Laws' men were struggling to break down a resolute defence and had keeper Lee Grant to thank for keeping the scoresheet blank.
Striker Marc Richards' through-ball found Louis Dodds but his effort was parried to safety by Grant with half-time approaching.
But Jeffers beat his team-mates to the changing room as fourth official David Coote alerted referee Graham Salisbury to an incident between Jeffers and Fraser for which he saw red.
The Valiants looked to capitalise on their numerical advantage after the interval but Wednesday keeper Grant denied Rob Taylor's goalbound header on the line.
But the deadlock was broken in the 63rd minute when Kris Taylor fired a magnificent 30-yard shot into the top left-hand corner from being found by Fraser from a corner.
And it got better for Vale three minutes later when Rob Taylor ran on to a Dodds cross and gave the keeper no chance to double the lead.
Laws threw men forward in a desperate attempt to fight their way back into the game but it was to no avail as Vale's defence held out.