The teacher, Jesmin Akther, has made no comment. Police said she went into hiding after parents complained.
Officials say 14 girls, aged between eight and 12, received burn injuries.
Bangladesh banned corporal punishment in educational institutions, including religious schools (madrassas), in 2010.
“Acting on a tip-off, we raided a house in the old part of Dhaka and we arrested Jesmin Akther, who is accused of burning the legs of her students. We are seeking to remand her in custody for seven days,” police officer Shafiqul Islam told the BBC’s Anbarasan Ethirajan in Dhaka.
The school has been temporarily closed following the incident and the “hellish experience” of the girls has been widely reported. Their injuries are not thought to be serious.
Bangladesh, which is overwhelmingly Muslim, has two types of madrassas.
There are more than 16,000 state-sponsored Alia madrassas across the country teaching more than five million students.
Apart from Islamic studies, students in these institutions learn English, maths and science.
The second type are Qaumi madrassas, which are independent and run by donations from people inside and outside Bangladesh. They focus mainly on Islamic studies.
The Talimul Koran Mahila madrassa falls into this category.