And they put the game to bed just three minutes later when Grant McCann finished off a neat move to dink the ball home.
It was a brave effort by Vale, who bow out of the competition having already beaten Sheffield United and city rivals Wednesday in the opening two rounds, with goalkeeper Chris Martin a stand-out performer.
Despite seeing Vale have a goal chalked off for offside in the opening two minutes, Scunthorpe soon found their feet and began to assert their authority.
Martin, who played such an integral role in the first-round win at Bramall Lane, was called into action to make a brave save with his feet after Jonathan Forte was sent through on goal.
The woodwork then saved Vale as, after McCann's wicked free-kick had been headed across goal, Hayes smashed his effort against the upright.
Scunthorpe were denied by the post for a second time inside the opening 20 minutes when Michael O'Connor created space for himself and unleashed a 25-yard drive that beat the sprawling Martin and crashed into the woodwork.
Despite losing their goalkeeper and skipper Joe Murphy to injury, the Championship side continued to pressure and the impressive Martin twice saved from Forte and again from McCann, while Marcus Williams also flashed a shot across goal as Scunthorpe searched for the breakthrough.
The second period began in similar fashion with Martin in the thick of the action. He comfortably saved an O'Connor drive before his reflexes were put to the test, palming away a Josh Wright effort from distance.
The Scunthorpe pressure was incessant but they could just not find a way through. O'Connor sent a curling free-kick inches wide and then McCann flashed a header wide under a heavy challenge from a Vale defender.
Martin produced another top drawer save to deny Matthew Sparrow and from the resulting corner Hayes headed agonisingly wide.
With extra time looming, Nigel Adkins' men had arguably their best chance to win it, but Jordan Spence, in space at the back post, side-footed O'Connor's corner well wide.
Then, with extra-time just two minutes old, Scunthorpe finally took the lead when Hayes fortuitously deflected O'Connor's shot past the wrong-footed Martin.
The lead was doubled just three minutes later when McCann advanced into the box after a one-two with Sparrow and he beat the onrushing Martin with a cool finish.
Vale rallied in search of a way back into the game but substitute Sam Slocombe denied Marc Richards and Robert Taylor with good saves as Scunthorpe held on.