Third round hero Shaq Coulthirst scored two in three minutes in the second half after Brentford took a 1-0 lead into the break after an Ollie Watkins 40th minute opener.
However, the Championship side hit back just shy of the hour mark as Neal Maupay scored from the spot before Sergi Canos put them back in front soon after.
There was another twist to the tale though as Dan Sparkes came off the bench to produce an outstanding free-kick with 15 minutes remaining, taking the tie to a replay.
The result means that our FA Cup dream lives to see another day.
It was to be a solid start to the game for Barnet who made three changes from the side who lost to Braintree in their last fixture, including a debut for goalkeeper Will Huffer.
The on-loan shot-stopper was called into action on a few occasions including the sixth minute when Maupay saw his effort saved on six minutes.
The wood-work was to thank soon after though when Yoann Barbet’s header from a corner crashed against the post and out of danger.
Huffer did well again to deny Josh Dasilva from distance before a brilliant curling strike from Moses Odubajo cannoned off the upright from 25 yards out.
It was Barnet’s turn to try and open the scoring on 26 minutes when man of the match Dan Sweeney’s header from a corner was dealt with by Daniels.
Odubajo then picked up a yellow card for a foul on Elito before Watkins’ tried his luck from distance and after an awkward bounce, found the bottom left corner.
Craig Robson saw his effort on the edge of the box fall wide before Ephron Mason-Clark and Jack Taylor failed to capitalise after a tackle on Coulthirst.
It was Elito’s turn to try and find an equaliser before the break but his shot whistled wide of the right post.
It was goals galore in the second half and it all started when Coulthirst fired home from close range five minutes after the break.
Three minutes later and our number 10 turned the game on it’s head, tapping the ball into the bottom corner, making it 2-1 to the North London side.
However, the celebrations were halted just before the hour mark when David Tutonda was adjudged to have brought down Watkins; the referee pointed to the spot.
Top scorer Maupay stepped up to take it and made no mistake, slotting the ball into the bottom left corner, Huffer diving to his right.
Coulthirst, on the hunt for a hat-trick, saw his header saved seven minutes later before hero-on-the-night Sparkes came on for Wes Fonguck shortly after.
However, Brentford took the lead again two minutes later after Sergi Canos was played through on goal before calmly striking low past Huffer.
However, the away celebrations were short-lived when Barnet were awarded a free-kick from 30 yards out.
Substitute Sparkes stepped up to take it before bending a delightful effort around the wall, finding the net via the crossbar in spectacular fashion.
The goal sent The Hive into frenzy but it was to be a very tense final 15 minutes.
Huffer was on hand again to deny Maupay with 10 minutes to go before producing another good save to keep out Said Benrahma’s effort from distance.
Odubajo then saw his low drive hit the post before Julian Jeanvier’s effort was blocked off the line following a corner.
Elito then made way for Andre Boucaud just shy of the five minutes of stoppage time in which neither side could find a winner, taking the tie to a replay.
6,215 spectators attended the game making it a record turn-out at The Hive London.
A date for the replay will be announced in due course with the winner of the tie travelling to Championship side Swansea City in round five.
Barnet side: Huffer, Alexander, Johnson, Robson ( c ), Coulthirst, Taylor, Fonguck (Sparkes), Elito (Boucaud), Tutonda, Sweeney, Mason-Clark.
Unused Subs: Matrevics, Harrison, Payne, Smith, Bettamer.
Attendance: 6215 (1822 Brentford)