Neil Moran and Steve Shea's side trailled to an early goal but Dave Freeman netted twice to send Sphinx ahead at the break.
In the second half, Declan Doherty scored a third for Sphinx with the visitors pullng one back to give Sphinx a 3-2 victory.
Sphinx now meet the winners of Stratford Town/Sutton Coldfield for a place in Round One against possible League One or Two opposition.
Sphinx's away game at Hednesford on Tuesday has been postoned due to Hednesford competing in the Birmingham Senior Cup.