The game was delayed by about 15 minutes, with hundreds of fans blocking the team bus from entering the stadium in protest of Chelsea's interest in forming the European Super League.
The hosts came into this fixture on the back of a morale-boosting victory over Manchester City that saw them book their spot in the final of the FA Cup.
Kepa Arrizabalaga made only his second league start under Thomas Tuchel, while Kai Havertz, Hakim Ziyech and Christian Pulisic were charged with leading the line.
A tepid first half saw both sides struggle to create chances, with Havertz going closest for Chelsea when his weak shot was comfortably saved by Robert Sanchez.
This was largely the story for the rest of the game, despite Thomas Tuchel bringing on Olivier Giroud, Timo Werner and Callum Hudson-Odoi to add more bite in attack.
Brighton were, however, the better side in the dying stages and hit the post late on through Danny Welbeck.
The visitors were reduced to 10 men in injury time when Ben White was shown a second yellow card by referee Stuart Atwell for a rash challenge on Hudson-Odoi.
The draw sees Chelsea climb up into 4th spot on the Premier League table, while Brighton remain in 16th place. Up next for the Blues will be a trip to West Ham. Meanwhile, the Sea Gulls will square off against Sheffield United on Saturday.
Here are five talking points from the fixture.
#5 Chelsea make life hard for themselves in the race for the top four
Chelsea came into Tuesday's game with a huge window of opportunity to enter the top four, with West Ham, Tottenham Hotspur and Liverpool all having drawn earlier in the matchday.
A victory for the Stamford Bridge side would have taken them one point clear of Leicester City into 3rd spot.
However, Chelsea faltered on home soil, making it consecutive games in which they failed to win at home against relegation-threatened opponents.
With just six games to go, Thomas Tuchel's side still have their destiny in their hands in the race for the top four.
However, a win here would have made their job much easier, considering that they still have marquee games against West Ham, Arsenal, Manchester City and Leicester City to come in their run-in.
#4 Impressive Brighton were deserving of all three points
Brighton and Hove Albion gave as good as they got against Chelsea, and they had the best chances of the game.
Despite having lower possession than Chelsea, Graham Potter's men had more shots in the game. They could have snatched all three points late on, but Welbeck's shot came off the post instead.