After a goalless first 45 minutes, Chris Wood put the visitors in front at Stamford Bridge with a cool finish after a well worked free-kick routine.
The hosts then applied more pressure and were level shortly after, with winger Noni Madueke levelling things up.
James Ward-Prowse was then sent off for Forest, before a melee involving almost all 22 players started close to the dugouts.
Chelsea missed a number of glaring opportunities in the closing stages, including a two efforts by Cole Palmer that prompted a superb double save by Mats Selz.
The draw leaves Enzo Maresca’s side fourth in the Premier League table, while Forest sit ninth.
Off the pitch, Chelsea appear to have been in talks about the potential signing of yet another Brazilian youngster.
According to CaughtOffside, both Chelsea and London rivals Arsenal are among a number of Premier League clubs that have shown interest in Santos teenager JP Chermont.
The 18-year-old is one of the most highly rated young players in Brazil and, following Santos’ relegation to the second tier, the club are believed to be open to the idea of selling him for less than his £58 million release clause.
Both the Blues and the Gunners have already engaged in talks with the player’s representatives, it is claimed, while the likes of Everton and Wolves have also been keeping tabs on the right-back.
Chermont is a product of Santos’ youth system and has now played 29 times for the club at senior level, scoring twice and registering three assists.
He is contracted to the club until the end of 2027, but the report claims that Santos could now sell the full-back in an effort to ease their financial woes following relegation.
