Hungerford had the chance to go 1-0 up from the penalty spot after just five minutes, but Luke Hopper’s penalty couldn’t find the back of the net.
Half an hour later Billericay made the breakthrough, Jake Robinson scoring against the run of play, giving the away side the advantage.
This got even worse when Alex Wall was sent off just 10 minutes into the second half.
Billericay had the chance to make it two, the Blues also receiving a penalty. Sam Deering took it but it was well saved by Crusaders keeper Marcus Beauchamp.
However, it was eventually two when substitute and former Arsenal man Sanchez Watt scored with a quarter of an hour left to play.
Defeat for Hungerford, while Billericay go on to face Havant and Waterlooville in the First Round Proper.
Elsewhere, Tom Day was named among the substitutes in Hemel Hempstead Town’s 1-1 draw at home against Bognor Regis Town, but he didn’t feature. The winner of the reply will face Taunton Town in the next round of the competition.