The incident happened at Fir Vale School on Owler Lane, Fir Vale shortly before 8.30am on Tuesday, February 13.
Pupils and staff escaped unharmed but pupils were sent home on the advice of South Yorkshire Police while an investigation was launched.
Two men in their 20s - one of whom was a former pupil - were arrested on suspicion of dangerous driving and causing criminal damage.
It was reported that the car was being driven 'erratically' around the school grounds for about 10 minutes before crashing into the building.
A message posted on the school's website yesterday afternoon said all parents and carers had been texted a link to a letter regarding the incident.
It added the school would reopen on Wednesday morning as normal.