Theo Richardson, pictured, saved two spot kicks as the Bulls saw off their fellow National League North side 4-2 after the tie finished 1-1.
The draw for the fourth round will take place at around 5pm on Monday, live on TalkSport 2.
It is understood that it won't be regionalised, so will be an open draw. Below are the teams Hereford could face.
Ebbsfleet
Fylde
Carlisle
Alvechurch
Anstey Nomads
Macclesfield
Clitheroe
Scunthorpe
Kidderminster
Marine
Rochdale
Southport
Tamworth
Halifax
Telford
Morecambe
Brackley
Leatherhead
Yeovil
Forest Green
Horsham
Chatham
Wealdstone
Bath
Woking
Dagenham
Eastleigh
Southend
Walton & Hersham
Gloucester
Hornchurch
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