Bromyard took the lead through McCarthy in the 34th minute but Sphinx were level through Kyle Rhodes in the 62nd minute.
Six minutes later, Kelvin Phillips put Sphinx ahead only for Townsend to level in the 76th minute.
Three minutes from time, Danny McSheffrey's side were awarded a penalty and Adie Fisher scored to set up a home Preliminary Round tie with Stourport Swifts on Saturday August 29th.
Elsewhere in the FA Cup, Stratford Town won 5-0 at home to Pilkington XXX.
Former Sphinx defender Adam Knibbs scored for Barwell in a 2-1 home defeat to Long Eaton whilst former Sphinx striker Tom Thacker scored twice for Highgate United in a 3-3 draw with Malvern Town and another former striker Adam Kinder scored for Coleshill Town in a 2-0 win against Bridgnorth.