Nicolas Jackson sent the Blues ahead in the 35th minute of the contest, before goals from Mads Roerslev and Yoane Wissa saw the Bees lead in the second period.
However, Disasi headed an equaliser for Mauricio Pochettino's side in the 83rd minute, as the points were shared in the entertaining West London derby.
The result has left Chelsea in 11th position in the table, boasting 36 points from their 26 league games this season, while Brentford sit 15th, six points outside of the relegation zone.
Smart defending from Roerslev prevented Enzo Fernandez from finding Jackson in a dangerous position outside the Brentford box in the early stages, before Disasi missed the target with a header moments later.
Brentford had a half-chance in the 10th minute when Sergio Reguilon's cross found Ivan Toney, but his shot was blocked by Chelsea defender Levi Colwill, which led to the home side's players calling for a penalty.
Wissa was next to try his luck for the Bees less than two minutes later, but the forward, who had broken into a dangerous position outside the Chelsea box, curled his effort just wide of the post.
Brentford had another opportunity in the 25th minute, with Toney delivering low into Wissa, but the latter's first-time effort was into the side-netting.
Jackson had a chance to send Chelsea ahead just past the 30-minute mark, but after rounding the Brentford goalkeeper, the striker's weak effort was blocked on the line by Mathias Jorgensen.
The breakthrough arrived for Chelsea in the 35th minute, though, with Jackson getting his name on the scoresheet, as the striker headed a brilliant delivery from Malo Gusto into the back of the net.
The Blues finished the first period well, with Ben Chilwell having a low shot blocked after smart work from Fernandez, before Jackson saw an attempt from a narrowing angle kept out by Mark Flekken, and Brentford were finding it difficult to get going after a bright start.
The hosts were bright at the start of the second half, with Vitaly Janelt having a long-range strike kept out, but Thomas Frank's side managed to level the scores in the 50th minute through Roerslev, who fired into the back of the net from close range after Chelsea failed to deal with a Wissa cross from the right.
Brentford came close to taking the lead shortly after, with Janelt striking from distance, but his effort came off the foot of the post in a let-off for Chelsea.
The visitors had two half-chances in the period that followed, but Cole Palmer missed the crossbar with a header, before Fernandez fired over the crossbar from distance.
Palmer had a glorious opportunity to make it 2-1 Chelsea just past the hour, but the forward's first-time effort from barely 12 yards out was wide of the post.
Wissa then missed the target with a header down the other end, before Reguilon fired over at the far post, with both teams threatening to score the game's third goal.
Brentford turned the match around in the 69th minute, and it was a sensational goal from Wissa, who found the back of the net with an excellent bicycle kick.
Reguilon headed against the post soon after, with the Bees coming close to scoring a third, which would surely have put the match beyond Chelsea.
The visitors managed to make it 2-2 in the 84th minute, though, with Disasi heading an excellent cross from Palmer into the back of the net from close range.
Chelsea were firmly on the front foot in the period that followed, with Flekken making a smart save to deny Raheem Sterling, with the West London derby set up for a fascinating conclusion, but neither could find a third in the final exchanges, as the points were shared at the Brentford Community Stadium.
Brentford will be bidding to build on the positive result when they head to Arsenal next Saturday, while Chelsea are next in league action at home to Newcastle United on March 11.