The Bees are runaway leaders in the Conference, but with boss Martin Allen selecting a predominantly inexperienced side, his men were blown away by their Conference South visitors after drawing 0-0 in their first game.
The hosts saw plenty of the ball during the opening stages but fell behind after eight minutes as Joe Gardner crossed for Steve Cawley to turn in from close range.
The FA Trophy
First Round Proper Replays
Tuesday 16 December
Winning clubs receive £5,000
Barnet drew level from the spot on 33 minutes through on-loan Watford man Bernard Mensah, but Sam Collins and Lewis Taaffe fired Concord into a 3-1 lead shortly after the break.
Once again Barnet hit back on 66 minutes with another penalty, young striker Mathew Stevens confidently dinking the ball home from 12 yards.
The comeback was not to be, however, as a rampant Rangers upped their attacking impetus and struck further blows through Steve Cawley, Mitchell Pinnock and Tony Stokes to make it six.
The Beachboys now face a trip to Hemel Hempstead Town in The Second Round.
Havant & Waterlooville were also in impressive goalscoring form as they hit five without reply past Weymouth.
Wrexham and North Ferriby United both enjoyed 2-0 wins against Southport and Boston United respectively, and Gosport Borough were 2-1 victors over Basingstoke Town after extra-time.
Forest Green Rovers won on the road by three goals to nil at Didcot Town while the fixture between Welling United and Ebbsfleet United was postponed.
