The store is one of 43 Tesco Homeplus centres across the UK to be shut because they failed to turn a profit.
Four of the outlets are in Scotland – in Kirkcaldy, Grangemouth, Troon and Edinburgh – and means 326 jobs will be lost.
The Fort Kinnaird store is due to close to customers on March 15 but local MP Sheila Gilmore has told how staff had “no idea” they would be out of a job in six weeks.
Announcing a series of closures, Dave Lewis, Tesco chief executive, said: “Earlier this month I announced that our performance as a business has fallen significantly short of where we would want it to be and that to protect the future of the business in the UK we would close 43 unprofitable stores.
“It is with great sadness that I have to tell you that our Edinburgh Homeplus store is one of the 43 we plan to close.
“The decision to close the store has been exceptionally difficult to take. I recognise it will affect many hard-working colleagues, our customers and the local community.
He added: “We will continue to do our best to serve them through our local stores in Edinburgh, Musselburgh and our dotcom service.”
Ms Gilmore said she was “devastated” for the 55 staff at Homeplus and has asked to be allowed to meet with them to discuss their futures.
She said: “Tesco kept workers waiting weeks for this news and I’ve been told that even [on Tuesday] staff had no idea that they would be out of a job so soon. They are going home worried about their families’ quality of life and their future.
“Tesco’s chief executive claimed in a letter to me that the store was ‘falling short’, but other than that they haven’t told me what the firm has done to increase trade and protect these jobs. I’m told that Tesco agreed a 25-year lease and a massive rent on the property as recently as 2007.”
The news comes as it emerged Tesco has recalled one of its own-brand squash drinks after customers complained of a “disgusting smell” and parents raised the possibility that it could have caused their children’s stomach upsets.
The supermarket said it had withdrawn the Tesco No Added Sugar Double Concentrate Apple and Blackcurrant 750ml and 1.5l products and was investigating complaints.