Watched by a Jenner Park crowd of 455, Barry's most recent FA Cup outing came against Weymouth in October 1992; and the Terras were victorious again on Saturday, winning 1-0 in Billericay to set up a meeting with Braintree.
Meanwhile, Willand Rovers, who drew 0-0 with Barry in the summer of 2009, have been handed a home draw against either Gosport Borough or Maidenhead United.
Willand, the lowest-ranked side still in the competition, secured their progress with an 3-2 win over Aveley Town at the weekend, after trailing at the interval.
Havant and Waterlooville, beaten at Jenner Park in 2001, will travel to Canvey Island, while frequent opposition of yesteryear Gloucester City host Forest Green Rovers; if Gloucester can vanquish Staines.
Near-neighbours across the water, Weston-Super-Mare, will try their luck at Witham Town, with Welsh representation provided by Wrexham (away at Macclesfield) and Colwyn Bay (home to North Ferriby, pending a win over Warrington).
The full Fourth Qualifying Round draw, as compiled by the Non-League Paper, can be found here.
Obstructed from entering the competition for more than two decades, Barry most recently made this stage in 1986-87; falling to current League One outfit Yeovil Town.
The Fourth Qualifying Round for 2014-15 will be held on the weekend of Saturday 25th October, with £12,500 and a potential tie against a Football League side awaiting for each winning club.
Pictured is former Jenner Park favourite Ian Love, on target for Barry in the 1984-85 FA Cup.