In a breathless first half, Brentford took the lead twice thanks to Pontus Jansson and Ivan Toney goals, but Brighton responded quickly to both, with Kaoru Mitoma and Danny Welbeck scoring equalisers.
An early second half strike from Ethan Pinnock looked like it was going to be enough for Thomas Frank's side to snatch the win in an encounter that was largely dominated by Roberto De Zerbi's men, but the hosts levelled late on thanks to Alexis Mac Allister's penalty.
The result means both clubs leapfrog Liverpool, with Brighton going sixth and Brentford into seventh, with the two sides tied on 43 points, and six points off an improbable top-four position.
With both clubs chasing European football for the first time in their history, the two started the day level on points with Liverpool, who lost earlier at Saturday lunchtime.
Brighton had the best of the early exchanges, with Mitoma causing trouble in behind forcing a corner out of Pinnock, which resulted in a major six-yard box scramble which Brentford just about survived.
With their first attack of the game though, the visitors took a 10th-minute lead, when Mathias Jensen's excellent cross saw Jansson power a header beyond Jason Steele in the Brighton goal.
The Bees were given a great chance to make it two just a minute later after Steele and Pervis Estupinan got into a mess and conceded an indirect free kick in their own box following a backpass.
The chance was squandered though, as with the entire Brighton team on the line, they managed to smother the opportunity as Jensen blasted the ball straight into the swarm of on-rushing defenders.
Brighton would go on to control the game, and pressure was mounting as Solly March was denied by David Raya, but just moments later Brentford would succumb to the Brighton onslaught.
It came via an unlikely provider in the form of goalkeeper Steele, whose fantastic ball right through the heart of the Brentford defence found Mitoma who calmly lobbed the ball over Raya who found himself in no man's land.
However, just seconds from the restart, Brighton were caught napping, with Lewis Dunk failing to get back in line with the rest of his defence following a throw in, allowing Bryan Mbeumo to flick on a forward pass into the path of Toney who controlled and finished clinically beyond Steele to put the visitors back in front.
It would not take long for the Seagulls to respond once again though, as March's dangerous delivery to the back post was met by the head of Welbeck, who beat Raya at his front post to make it 2-2 inside half an hour.
Raya and the Brentford defence were kept busy for the rest of the first half, as March and Levi Colwill both went close shortly before half time.
Some more magic from Mitoma caused chaos in the opposition area once more, and after a few ricochets, Welbeck was unfortunate to be in a position in the six-yard box that blocked the ball from going over the line.
The second half action picked up right where it left off in the first 45, and it was Brentford who would once again score against the run of play.
A brilliant set piece from Mbeumo allowed Pinnock to spin off round the back post and show the composure of a seasoned attacker to delicately touch the ball beyond Steele with the outside of his left foot to put Brentford 3-2 up.
After overplaying in their own penalty area, Brighton were fortunate to not concede a fourth, as Toney was presented with a glaring chance, but he fired straight at Steele, who was out of position, with the goal at his mercy.
From there on out it was all Brighton as Raya's goal came under siege from the hosts who frantically searched for a third equaliser.
Moises Caicedo forced the Spanish goalkeeper into a smart save to tip his powerful strike over the crossbar, before Pascal Gross and Dunk both failed to convert with excellent headed opportunities.
In an enthralling encounter from start to finish, there would be late drama too, as Aaron Hickey was penalised for a handball near the goal-line, giving Brighton a late chance to draw level from the penalty spot.
Hickey was potentially fortunate to avoid a red card, but Mac Allister made no mistake, netting his fifth penalty of the season to score the sixth goal of the afternoon and a remarkable third equaliser for Brighton.
Despite six minutes of added time, neither side managed to create any more meaningful opportunities as the spoils were shared in one of the games of the season.
Brighton travel along the South Coast to face Bournemouth in midweek in one of their many games in hand, while Brentford have a tough trip to Old Trafford to face Manchester United on Wednesday in their next encounter.