Students were seen jumping and falling from the commercial building in the western city of Surat as black smoke billowed from windows.
The fire is believed to have been started by an electrical fault in the air conditioning, officials said.
Police said one of the owners of the tuition centre had been arrested.
Surat police chief Satish Sharma told reporters that three people in total had been charged with culpable homicide and that the two others would be arrested soon. As anger mounted in the local community, he appealed to people to remain calm and to not take the law into their own hands.
At least 20 other people suffered serious injuries in the fire and were being treated in hospital in Gujarat.
“The students lost their life both because of the fire and jumping out of the building,” Deepak Sapthaley, a local fire official, told AFP.
Surat’s fire and emergency services commissioner U Upadhyay said: “The fire was caused by a short-circuit in one of the air-conditioning ducts on the ground floor… and spread to the third floor.”
All of the dead were aged below 20 years and many were trapped because the fire began near a staircase, Reuters reported.