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 vs. West Brom
Peterborough vs. Fleetwood/Leicester
Huddersfield vs. Birmingham
Notts County vs. Wolves/Swansea
Yeovil vs. Manchester United
Carlisle/Sheffield Wednesday vs. Stevenage/Reading
Cardiff/Mansfield vs. Manchester City
MK Dons vs. Coventry
Millwall vs. Rochdale
Southampton vs. Watford
Middlesbrough vs. Brighton/Crystal Palace
Bournemouth/Wigan vs. Shrewsbury/West Ham
Hull vs. Nottingham Forest
Newport County vs. Tottenham
Norwich/Chelsea vs. Newcastle
Sheffield United vs. Preston