Shortwood United, the lowest ranked team left in the FA Cup, will host Port Vale in the first round after winning their replay at Aldershot.
The Southern League Division One South & West side beat the Conference Premier outfit 2-1 on Tuesday.
Daventry Town, also in the eighth tier, have been drawn away at Chesterfield.
There will be a Lincolnshire derby as Conference Premier side Grimsby host Scunthorpe United, of League Two, in the ties played on 9 and 10 November.
AFC Wimbledon, the phoenix club set up by supporters after the demise of 1988 FA Cup winners Wimbledon, will welcome Coventry City, the side that lifted the famous trophy a year previously.
Conference South side Bishop's Stortford will welcome Aidy Boothroyd's League Two strugglers Northampton Town, while their league rivals Staines Town travel to near neighbours Brentford, where Swans manager Marcus Gayle started his career.
And fellow non-league clubs Gloucester and St Albans City host Football League opposition in the form of Fleetwood and Mansfield respectively.
League One leaders Leyton Orient, the highest ranked side to come in at the first-round stage, have a home tie against Conference Premier Southport.
There are four all League One ties with a Yorkshire derby between Rotherham and Bradford City, Walsall taking on Shrewsbury, Wolves travelling to Oldham and Sheffield United going to Colchester.
FA CUP FIRST ROUND DRAW
Morecambe v Southend United
Walsall v Shrewsbury
Boreham Wood v Carlisle United
St Albans City v Mansfield Town
MK Dons v FC Halifax Town
Bristol Rovers v York City
Tamworth v Cheltenham Town
Grimsby v Scunthorpe United
Bury v Cambridge United
Corby Town v Dover Athletic
Colchester United v Sheffield United
Oxford United v Gateshead
Hartlepool United v Notts County
Wrexham v Alfreton Town
Chesterfield v Daventry Town
Bristol City v Dagenham & Redbridge
AFC Wimbledon v Coventry City
Preston North End v Barnet
Bishop's Stortford v Northampton Town
Burton Albion v Hereford United
Rotherham United v Bradford City
Gillingham v Brackley
Salisbury v Dartford
Accrington v Tranmere Rovers
Brentford v Staines Town
Stevenage v Portsmouth
Shortwood United v Port Vale
Leyton Orient v Southport
Peterborough United v Exeter City
Torquay United v Rochdale
Braintree Town v Newport County
Oldham Athletic v Wolves
Lincoln City v Plymouth Argyle
Kidderminster Harriers v Sutton United
Hednesford Town v Crawley Town
Stourbridge v Biggleswade Town
Welling United v Luton Town
Macclesfield Town v Swindon Town
Wycombe Wanderers v Crewe Alexandra
Gloucester City v Fleetwood Town