Only the facade will remain of the Tap and Tankard and the former Chubbys takeaway at 24-26 Cambridge Street.
A new three-storey building including a cafe, restaurant, bar, retail and offices will be built behind.
A planning committee heard that the unlisted building could not be adapted to modern standards, which Historic England confirmed.
Historians said the building should be preserved because of its links to neighbouring Leah’s Yard, the grade II listed Little Mesters’ workshop it was built to serve.
Why is the Tap and Tankard important to Sheffield’s heritage?
Robin Hughes, of Joined Up Heritage Sheffield, said: “Sheffield pubs were integral throughout our world famous industries, with dust causing respiratory illness and intense heat causing dehydration.
“Water lacked essential mineral content and might be unwholesome so beer is as much part of the Sheffield story as coal or crucibles. With Leah’s Yard we have a factory and a pub built together at the same time, on the same plot, the only such buildings known to be still standing in Sheffield.
“So this is not a plea to save an old pub but a collection of the domestics workshops which the Leah’s Yard pub makes unique. Leah’s Yard itself is already refurbished and is safe so we need to follow through and not spoil what’s been achieved so far.
“The history of this great city is as much about our people as it is about industry. We value our industrial buildings because they recorded people’s trades. What is uniquely special about Leah’s Yard pub is it’s a record of people’s lives.”
What did Sheffield councillors think about the plans?
Nine councillors voted in favour of the demolition, with two against and one abstention.
Coun Peter Price said: “When I first heard of demolishing this pub and Chubbys I fully intended to oppose it but having visited, I’m very excited about what’s happening. I have fond memories of being in Chubbys in the early hours of the morning and being in the pub, but it really is virtually impossible to adapt that pub to modern standards.
“It’s very uneven, the stairs are very dangerous and it’s not adaptable to modern standards. Preserving the frontage is ideal because it fits in well with the rest of Cambridge Street.”
Coun Andrew Sangar agreed: “I found this one really difficult. Mr Hughes made a really good case in terms of the history, how few Little Mesters sites we’ve got and the integrity of the pub.
“But I don’t think you’ll convince me that is the original 1863 interior and that the pub has not been touched. I think the reality is, it’s had several refurbishments over the years.
“This is an exciting scheme, Leah’s Yard is going to be protected whatever but this balances the history of the site with a new focal point for Heart of the City 2.”