The government promised to appeal for overseas aid for schools
Teachers in Zimbabwe have called off a strike despite their wage demands not being met, while the government has slashed school fees for the new term.
David Coltart said the government had no money to raise their salaries, but he had agreed to help teachers by giving their children free schooling.
Teachers' groups said they accepted the government was struggling for funds and needed time to raise revenue.
Teachers are paid $100 (£66) a month but unions wanted four times as much.
Mr Coltart has been in protracted talks with unions and foreign aid donors to make sure the schools reopened in time for the new term on Tuesday.
The Zimbabwean government has also met unions' demand for a huge cut in school fees - which most parents cannot afford - to get children back into the classroom.
Zimbabwe's state education system had virtually collapsed until the new power-sharing government agreed to pay teachers in foreign currency in February.
Mr Coltart said that although the new term would begin, the education system was a "shadow" of what it had been.
"The doors may open, there may be children in the classrooms and teachers teaching, but there are very few textbooks in the rural areas and many schools do not have roofs or doors or windows," he told the BBC.
Mr Coltart, a former opposition activist, said the state of the service was down to two decades of neglect by President Robert Mugabe's government.
Raymond Majongwe, head of the Progressive Teachers' Union, said going back to work was the "responsible" thing to do, even though all their demands had not been met.
He told the BBC's Network Africa programme: "We have no reason to proceed with the strike action that will do nothing but confuse the situation that we are trying to ultimately address.
"As the government does not have the capacity to address the problems that it faces, so the donors need to chip in."
He said they were pleased Zimbabwe's government had agreed to cut school fees from between $50 (£33) and $150 (£100) a term to a maximum of $20 (£13) and that teachers' children would be exempt from fees.
Most Western donors, however, remain reluctant to restore aid to Zimbabwe while Mr Mugabe remains president.
Last week, African countries agreed to give Zimbabwe $400m-worth of credit.
Teachers and other public sector workers agreed to end industrial action in February after the government agreed to pay their salaries in foreign currency because of massive inflation.
Zimbabwe's economic collapse had left the country's currency virtually worthless - and many civil servants unable to afford even the bus fare to work.
The coalition government formed in February between President Mugabe and Prime Minister Morgan Tsvangirai, leader of the Movement for Democratic Change, faces calls from eviscerated public services across the spectrum for more funding.
On Friday, Mr Tsvangirai told a Labour Day rally that the government was broke and could not afford to pay any more than the allowance of $100 a month.
He said this was also what Mr Mugabe was being paid.