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.
#3 Did the European Super League hullabaloo distract Chelsea?
The footballing space has been dominated by news of the European Super League in the last 48 hours. Chelsea found themselves in the middle of the storm, given their role as one of the founding clubs.
This led to fans staging a protest in front of the stadium for several minutes, with Petr Cech having to personally intervene and plead with the fans to allow the team bus to move ahead.
Thomas Tuchel addressed this in his post-match press conference and revealed that he, alongside his players, was stressed from the events of the last few days.
This does not excuse what was a dire performance from Chelsea, with their seven shots across 90 minutes representing the lowest number of shots they have attempted in a Premier League game under Thomas Tuchel.
#2 Defensive solidity on full display
Chelsea's defense under Thomas Tuchel has been nearly impenetrable, with this shutout against Brighton marking the 15th clean sheet they have kept in 20 games under the German tactician.
This defensive solidity has been behind the gains made since the turn of the year, and this is far from the side that struggled to keep opponents out in the recent past.
#1 Attacking woes in front of goal continues for Chelsea
For all their defensive solidity, the west London giants have looked extremely blunt in attack, and their struggles came to the fore against the Sea Gulls.
The current Chelsea attack is littered with quality names on paper, including Timo Werner and Hakim Ziyech, but they have massively failed to live up to expectations.
Against Brighton, they struggled to get going in attack and failed to test Sanchez. Tuchel made three changes in attack in the second half, but that failed to make any noticeable impact in the game.
This makes it four consecutive games in which the Blues have failed to score more than a goal, with just nine games under Tuchel seeing Chelsea score two or more goals.
Despite the gains made overall, Thomas Tuchel has to find a way to solve the problems facing Chelsea in front of goal, otherwise, they run the risk of faltering at the final hurdle of the season.