The store had been due to close under a restructuring plan that House of Fraser announced in June.
However, the department store group fell into administration earlier this month, and was bought by Mike Ashley's Sports Direct.
The deal is the first covering a House of Fraser store since the takeover.
The store was one of the 31 earmarked for closure under the CVA restructuring deal that was first struck before Mr Ashley stepped in with his £90m rescue bid.
His plan is to keep 80% of House of Fraser's 59 stores across the UK open. A spokesperson said they were working through the list on "a store-by-store basis".
Property giant CBRE, which is advising Sports Direct, said negotiations over the Oxford Street outlet had been conducted at "great speed".
James Keany from CBRE said: "This deal only happened because all parties realised it was better to keep the store open and fully operational.
"It was a real case of landlord and tenant genuinely working together and at great speed. Everyone was sensible about the terms of the transaction."