The League Two outfit overcome Bradford City 2-0 on Saturday afternoon to keep their dream alive and will now welcome the 12-time winners to Huish Park later this month.
Elsewhere in the fourth-round draw, Liverpool have been drawn at home to West Bromwich Albion, while Chelsea - who first face a replay against Norwich City - face the prospect of hosting Newcastle United.
Runaway Premier League leaders Manchester City will head to either Cardiff City or Mansfield Town, meanwhile, with the two sides facing off for a second time next week following their goalless draw at the weekend.
FA Cup fourth-round draw in full:
Liverpool v West Brom
Peterborough v Fleetwood/Leicester
Huddersfield v Birmingham
Notts County v Wolves/Swansea
Yeovil v Manchester United
Carlisle/Sheffield Wednesday v Stevenage/Reading
Cardiff/Mansfield v Manchester City
MK Dons v Coventry
Millwall v Rochdale
Southampton v Watford
Middlesbrough v Brighton/Crystal Palace
Bournemouth/Wigan v Shrewsbury/West Ham
Hull v Nottingham Forest
Newport County v Tottenham
Norwich/Chelsea v Newcastle
Sheffield United v Preston