The two sides drew 2-2 a week ago and needed a replay to separate them as they looked to decide who would face Barnsley-based Shaw Lane Aquaforce in the last eight.
Dunston 1-3 Glossop NE AET
The FA Vase
Fifth Round
Saturday 21 February
And the hosts hit the front early on through Gary Ormston, whose effort took a huge deflection to wrongfoot visiting goalkeeper Greg Hall.
Glossop were building the pressure, though, coming close a number of times before the interval with home goalkeeper Liam Connell in great form as he denied Dave Young and Tom Bailey.
But there was nothing Connell could do when Glossop defender Matt Russell fired in a superb effort from distance that left the goalkeeper with no chance.
Neither side could find a winner in normal time but Glossop scored twice in the first-half of extra-time to take a decisive step towards the Sixth Round.
Captain Young grabbed the first, smashing in to give Glossop the lead before forward Kieran Lungsden made it 3-1 to ensure they progressed at Dunston's expense.
