Members of the Public and Commercial Services Union will walk out on 19 May.
In a ballot, 96% of the 150 affected members voted for strikes on a 53% turnout. 247 people are employed at the office in St Paul's Place.
BIS said it will provide "comprehensive support" to affected staff.
The government announced plans to shut the office by 2018 in January.
Mark Serwotka, general secretary of the Public and Commercial Services Union, said: "This vote sends the clearest possible message to BIS ministers and senior officials that these plans are wrong and must be reversed.
"The closure of this office is precisely the opposite of what the government claims to want for the North and the wider civil service."
He said the union was also considering a timetable for further walkouts.
Announcing the plans, permanent secretary Martin Donnelly said it was part of a move to create a combined central HQ and policy centre in London and establish six business centres across the UK.
He said the decision was taken in an effort to "modernise the way we work, reduce operating costs and deliver a simpler, smaller department that is more flexible and responsive".
A BIS spokesperson said: "Whenever a proposal such as this is made, people's livelihoods and families are directly affected, and these decisions are never easy. It remains our top priority that all our staff are fully briefed and consulted during the process.
"We take their future and the contribution they have made very seriously, and will provide comprehensive support to all those facing a potential change or loss of job."
Civil servants at the Department for Business, Innovation and Skills (BIS) have announced plans to stage a 24-hour strike in protest at plans to shut the department's Sheffield office.