The League Two outfit took the sting out of the game and the excellent atmosphere at the Plant Hire Community Stadium by taking an 8th minute lead when Josh Gordon fired across Puritans `keeper Ben Taylor.
The hosts held the Bluebirds to that one goal lead until the break but had few clearcut openings themselves.
And 10 minutes into the second half Banbury fell further behind when, shortly after Taylor had produced a wonderful save to deny Jordan Stevens, Offrande Zanzala made it 2-0 following a defensive mix-up.
Banbury then had their best spell of the game but again were unable to make any clear opportunities.
And the tie was over for them in the 80th minute when Taylor brought down Zanzala and Ollie Banks converted the resulting penalty.
Stevens rubbed salt into Banbury`s wounds when he made it 4-0 in the 83rd minute.
Banbury`s fine run comes to an end, and it`s now left to league rivals Stratford Town to carry the flag in tomorrow`s tie against Shrewsbury Town.
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