Steven Schumacher’s side reached this stage with a 2-1 victory against League One rivals Huddersfield Town at the weekend thanks to goals from Ibrahim Cissoko and Richard Taylor either side of half time.
Meanwhile, Swindon won 2-1 at Rotherham United in extra time after Darren Oldaker’s late winner at the New York Stadium.
Swindon have had a strong start to the campaign under Ian Holloway, currently sitting second in the League Two table and only behind leaders Walsall on goal difference.
Wanderers’ last trip to the County Ground was in October 2016 in League One and Josh Vela scored a late winner with the help of a deflection.
The draw was carried out by ex-Manchester City defender Joleon Lescott and former Fulham midfielder Steve Sidwell live on TNT Sports in the build-up to Tamworth’s tie against Leyton Orient.
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Second-round ties will take place across the weekend of December 6, between Wanderers’ League One trips to Luton Town and Mansfield Town.
Full FA Cup second-round draw
Slough Town v Macclesfield
Grimsby Town v Wealdstone
Sutton United v Shrewsbury Town
Swindon Town v Wanderers
Chelmsford City v Weston-Super-Mare
Peterborough United v Barnsley
Boreham Wood v Newport County
MK Dons v Oldham Athletic
Wigan Athletic v Barrow
Fleetwood Town v Luton Town
Salford City v Tamworth or Leyton Orient
Accrington Stanley v Mansfield Town
Stockport County v Cambridge United
Brackley Town v Burton Albion
Blackpool v Carlisle United
Gateshead v Walsall
Exeter City v Wycombe Wanderers
Cheltenham Town v Buxton
Chesterfield v Doncaster Rovers
Port Vale v Bristol Rovers