Thailand lost 5-2 after extra time on Tuesday, ending the game in the Cambodian capital Phnom Penh with eight players on the field.
One of those sent off was Thai goalkeeper Soponwit Rakyart after he ran half the length of the pitch to join in the second brawl, delivering a diving punch to an opponent.
"The FA of Thailand must apologize for the clash on the touchline," the body said in a statement, adding it would "set up a committee to investigate those involved as soon as possible and will take decisive measures".
Men's football at the biennial regional event is played between under-23 sides.
The final had been billed as a chance for Indonesia to restore some pride to its football following a deadly stadium disaster and the loss of hosting the Under-20 World Cup.
While the team achieved that, the game will be more widely remembered for the scenes which began in the 97th minute when Thailand, who had been 2-0 down, scored to make it 2-2 and force extra time.
Thai officials celebrated their late leveler by running over to the Indonesia bench, prompting the first melee.
When Indonesia took the lead back early in extra time, their officials returned the favor, with even more incendiary results.
Punches and kicks were landed, and security officers had to intervene. Both teams had a player sent off on that occasion, and members of the coaching staff were also dismissed.
As the Thai team's discipline crumbled, another two of their players were sent off during extra time for second yellow cards.