The Wiltshire Robins currently lie ninth in the division above their Gloucestershire counterparts, making it a tough draw for Mark Yates' side. The two clubs have never met in the FA Cup before.
Chairman Paul Baker said: "Supporters will I am sure have mixed feelings about the FA Cup draw but personally I think it is a great draw.
"Not only are we at home, and our home record has been so much better this season, but we entertain arguably our biggest local rivals and fellow 'Robins' Swindon Town.
"It promises to be a fantastic afternoon, there is bound to be a big following from Wiltshire and I am sure Cheltenham Town fans will relish the opportunity of knocking out our near neighbours. Bring it on!"
The tie will be played on the weekend of November 8-9.
There is £18,000 at stake for the winners as well as a place in the second round of the competition.
Full first round draw:
Forest Green v Scunthorpe
Blyth Spartans v Altrincham
Canvey or Havant and Waterlooville v Preston
Oldham v Leyton Orient
Coventry v Worcester City
Hartlepool v East Thurrock United
Barnsley v Burton
Plymouth v AFC Fylde
Macclesfield or Wrexham v Woking
Port Vale v MK Dons
Eastbourne or Dover v Morecambe
Northampton v Rochdale
Cambridge Utd v Fleetwood
Bury v Nuneaton or Hemel Hempstead
Stevenage v Maidstone Utd
Luton Town v Newport
Southend v Chester FC
Tranmere v Bristol Rovers
Grimsby v Oxford Utd
Crewe v Sheffield United
Norton or Shildon v Gateshead
Portsmouth v Aldershot
Gillingham v Bristol City
Chelmsford or Barnet v Wycombe
Weymouth or Braintree v Chesterfield
Bromley v Dartford
Cheltenham v Swindon
Peterborough v Carlisle
Weston-super-Mare v Doncaster
Dag & Red v Tamworth or Southport
Basingstoke or Harrow v Spennymoor or AFC Telford
Eastleigh v Alfreton or Lincoln City
Warrington v Exeter City
Chorley or FC Halifax v Bradford City
Notts County v Accrington Stanley
Walsall v Shrewsbury
York City v AFC Wimbledon
Yeovil v Crawley Town
Gosport Borough v Colchester United
Mansfield Town v Concord Rangers