The defender pounced twice in the second half at Bramall Lane to ensure it was the League Two club, not David Weir's side, who progressed.
United captain Michael Doyle had earlier levelled with a spectacular finish. However, just as extra time was looming, Hussey claimed his second of the evening deep into injury time to stun the home crowd.
United seized control from the first whistle and created numerous opportunities before falling behind against the run of play.
Jordan Pickford, Albion's on-loan goalkeeper, nearly embarassed himself collecting a routine throughball but regained his composure to twice deny Chris Porter with acrobatic saves.
Darryl Westlake flashed wide from an acute angle as United turned the screw but Adam McGurk, making his debut for the visitors, had earlier served notice that Gary Rowett's team still posed a threat when he tested Mark Howard after meeting Hussey's cross.
The defender edged Albion in front five minutes after the restart when United failed to deal with McGurk's centre.
But their lead lasted less than 15 minutes as Doyle converted in superb fashion from 25 yards.
Extra-time beckoned but Hussey powered home an unstoppable header late on after meeting substitute Alex MacDonald's cross.
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