The Ryman Premier League side stormed to a surprise first-half lead when Adrian Toppin headed home before Matty McNeil sent the tie to extra-time slotting home from close range late on.
But Staines, who beat Brackley on penalties in the first qualifying round, came up trumps with Allaway denying Anthony Pilkington and David Poole from the spot to set up a second round clash with Peterborough.
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