Emma Hayes’ side, who started the day four points behind the Gunners, dominated possession but could not find a breakthrough from their 26 attempts.
Brighton withstood the pressure and defended resolutely, with goalkeeper Megan Walsh making a number of important saves.
Erin Cuthbert also hit the crossbar to sum up Chelsea’s frustration.
Following a run in which they scored in 59 consecutive WSL matches, this is the first time Chelsea have failed to score in consecutive league games since October 2018.
Despite making it seven successive games without scoring themselves, the result means Brighton put an end to their run of consecutive defeats.
The first real chance of the game fell to the hosts, with Danielle Carter’s long-rage effort drawing an impressive save from Ann-Katrin Berger.
Chelsea were playing without the WSL’s top scorer Sam Kerr, who is on international duty with Australia at the Women’s Asian Cup.
They created plenty of chances and came closest to an opener on 25 minutes when Pernille Harder made a clever run in behind the Brighton defence, but both Fran Kirby and Guro Reiten could not get enough on the end of her cross at the far post.
Kirby was through on goal and had a chance to make amends late in the first half, but she side-footed her effort too early allowing Walsh to make the low save.
That wasted chance summed up the Blues’ first half, they dominated possession but managed only two shots on target from their 10 attempts on goal.
It was much the same after the break, Kirby twice went close before an unmarked Erin Cuthbert blasted her effort wide of Walsh’s post from inside the box.
Cuthbert was left with her head in her hands not long after when she thought Chelsea had finally found a breakthrough, but her powerful shot from inside the box ricocheted off the crossbar.
Beth England had another big chance to win it late on for the reigning champions, but her downward header from Harder’s cross was once again saved by Walsh.
After Manchester United’s win over Tottenham, Chelsea slip to third in the WSL, three points behind Arsenal who take on Manchester City at 18:45 GMT.