David Tutonda opened the scoring in the early stages before Jack Taylor made it two just past the 15 minute mark.
Barnet looked to have taken a two-goal lead into half-time but Reece Grant pulled one back for the visitors moments before the break.
Lyle Della-Verde levelled the scores on 66 minutes before Elliott Johnson marked his return from injury with a goal to regain Barnet’s lead.
The outcome was settled six minutes from time when Cheye Alexander’s first goal for the club guaranteed progression to Round One.
John Still named an unchanged side that beat Solihull 2-0 in their previous encounter two weeks ago. Dave Tarpey joined goal scorer Johnson on the bench following spells out with injury.
A scorching afternoon in North London saw The Bees get off to a flyer after taking the lead with just six minutes on the clock. Josh Walker fizzed in a low ball across goal from the right which missed Manny Duku but Tutonda was on hand to smash home at the far post.
However, The Iron could have bagged themselves an equaliser soon after when a Jon Muleba cross was cleared out to Cameron James whose strike from distance whistled over the bar.
The missed chance proved costly as The Bees went up the other end and doubled their advantage. After Duku’s blocked effort fell to Taylor, the Barnet number 12 made no mistake from close range and nestled the back of the net.
Moments later, Walker bombed down the left before cutting inside and letting rip from inside the area but his effort was well saved by Ben Killip who dived to his near post.
Braintree’s Kodi Lyons-Foster picked up the first booking of the game after a foul on Tutonda but when Taylor struck the resulting free-kick, his attempt fell just wide of the post.
The visitors had a chance to pull one back on 36 minutes when Grant was played through on goal. However, his first-time shot from the edge of the box was well saved by Mark Cousins.
Braintree’s number 25 did find his name on the score-sheet on the stroke of half-time though as he scrambled the ball over the line to half the deficit going into the break.
Charlee Adams found Cheye Alexander 13 minutes into the second period but after the full-back cut inside and let fly from the edge of the area, his effort could only find a Braintree defender.
Up the other end and Grant had a real chance to net his brace. After he was played through on goal, his half-volley forced Cousins into a spectacular save to maintain his side’s lead.
However, The Iron did find an equaliser soon after as Muleba’s cross from the right found Della-Verde who volleyed the ball home from close range.
The Bees tried to regain their advantage minutes later but Adams’ strike from distance took a huge deflection, flying wide of the post.
With just over 15 minutes remaining, substitute Jack Barham missed a golden opportunity to put the hosts back in front after failing to convert a low Taylor cross from close range.
However, Barnet did regain their lead on 79 minutes when Alexander floated in a delightful ball before substitute Johnson nodded in at the far post.
The Bees made sure of their place in the next round after Alexander’s low free-kick from outside the box beat Killip in the Braintree net.
Barnet now have to wait until Monday’s draw to find out who they will be playing in Round One.
Barnet Line-up: Cousins, Alexander, Robson (C), Adams, J.Taylor (Mason-Clark), Fonguck, Walker, Tutonda (Johnson), Sweeney, Jules, Duku (Barham).
Unused Subs: Matrevics, Tarpey, Sparkes, Elito.
Attendance: 1057 (156 visitors)
