Paul Doswell's side will start as hot favourites to progress against the Kent side who play three levels below them in the Southern Counties East.
Chatham are one of only five step 5 clubs - the same level as the Premier Division of the Wessex League - left in the competition.
Two of them - Longridge and Skelmersdale - have been drawn against each other while Wessex Leaguers Christchurch host Hawks’ National League South rivals Dulwich Hamlet and Sheppey United travel to Eastbourne Borough.
Doswell, however, believes the tie is nowhere near as straightforward as some might think.
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'It is a home draw which is what you want,' he said.
'But I wouldn't take any notice of where they are in the football pyramid.
'A number of their team played for Dartford last season and they have got some good experience in their side.
'Obviously they have got some heavy backing into the club in a bid to win their league.
'They also won against a good Isthmian Premier side, Folkestone Invicta, in the last round.
'We certainly won't be underestimating them because of the quality they have in their side.'
Chatham have already played three games to reach this stage, beating Westside away (2-0), Southall home (3-2) and Folkestone Invicta away (3-0).
They have also won all five of their Southern Counties East games, scoring 26 goals and conceding six.
Jack Evans was last season’s top scorer with 26 goals in all competitions, and he already has eight in six matches in 2020/21.
Dan Bradshaw, who netted twice at Folkestone, has six in five appearances following his summer move from Sheppey United.
No stranger to goals, Bradshaw struck 45 times for Sheppey in 2019/20 even though the season was cut short.
Chatham are an old club boasting a proud history and way back in 1888-89 reached the quarter finals of the FA Cup where they lost 10-1 against West Bromwich Albion.
Hawks v Chatham will be played on Tuesday, October 13 - behind closed doors, sadly - with the winners picking up £5,625 and the losing club receiving £1,875.
Last season’s FA Cup non-league darlings Chichester City have been handed a third qualifying round home tie against two tiers higher Tonbridge Angels.
Third qualifying round draw: Bristol Manor Farm v Cray Wanderers, Chichester City v Tonbridge Angels, Cray Valley v Aveley, Eastbourne Borough v Sheppey, Ebbsfleet United v Chippenham Town, Hawks v Chatham, Wimborne Town v Maidstone, Southport v South Shields, Longridge Town v Skelmersdale United, Chester v Spennymoor Town, Marske United v Blyth Spartans, Farsley Celtic v AFC Fylde, Darlington v Tadcaster Albion, Marine v Nantwich Town, Chorley v York City, Guiseley v Matlock Town, FC United of Manchester v Alfreton Town, St Albans v Mickleover, Braintree Town v Maldon & Tiptree, Ilkeston Town v Alvechurch, Bishop's Stortford v Royston Town, Stafford Rangers v Hereford, Peterborough Sports v Banbury United, Leiston v AFC Telford United, Brackley Town v Kettering Town, Cambridge City v Halesowen Town, Oxford City v Tamworth, Bury Town v Nuneaton Borough, Boston United v Hemel Hempstead, Haringey Borough v Bracknell Town, Bedfont Sports Club v Canvey Island, Hartley Wintney v Barking, Hayes & Yeading v Chipstead, Slough Town v Bath City, Taunton Town v Truro City, Christchurch v Dulwich Hamlet, Sholing v Walton Casuals, Weston-super-Mare v Larkhall Athletic, Hampton & Richmond v Hornchurch, Burnham or Whyteleafe v Concord Rangers.