Nigel Starkey's side led Bromsgrove Rovers 2-0 before a goal reduced their lead.
Bromsgrove converted a penalty just before the end of the second half to send the game into extra time.
Sphinx went back in front before another penalty awarded by referee Andrew Stickley, made it 3-3.
Sphinx hit a fourth in the 117th minute and held on to book their place in the next round.
