Both clubs were charged with failing to control their players and officials while Forest striker Nathan Tyson was charged with improper conduct.
Tyson had paraded in front of Derby supporters waving a corner flag following his side's 3-2 victory, which sparked a melee involving players and staff.
Derby and Tyson admitted the charges. Forest denied their charge but an FA Regulatory Commission found it proved at Thursday's hearing.
Derby were warned as to their future conduct and fined £20,000, £15,000 of which is suspended.
Forest were warned as to their future conduct and fined £25,000, £10,000 of which is suspended.
Tyson was warned as to his future conduct, fined £5,000 and given a suspended two-match ban.
All suspended sanctions will be activated if the parties are found guilty of similar offences before the end of the 2010-11 season.
Commission chairman Peter Powell stated: "The commission found this incident to be both unsavoury and avoidable, which was caused in the outset by the actions of a professional footballer who, on this occasion, behaved naively and below the level that would have been expected.
"Such action, particularly in relation to a high-profile local derby, caused a safety issue to the general public which simply should not have happened.
"The regulatory commission however recognised the subsequent immediate acceptance of the charge, as they did in relation to the acceptance of the charges by Derby.
"The commission recognised the exemplary record of both clubs, which is the main reason for part of the sanctions being suspended, and would hope that both clubs do all within their power to ensure no such incident is repeated in the future."