May Brown from Weymouth, Dorset was told her sister Martha was a "10 out of 10" tissue match.
Martha was initially refused a visa to the UK because her income was too low.
The Home Office said it reversed its decision on her visa application due to "exceptional circumstances".
It comes after more than 61,000 people signed a petition against the refusal.
Immigration Minister Robert Goodwill said: "I have carefully considered the case of May Brown and decided that her sister will be granted leave to enter the UK given the compassionate and exceptional circumstances."
Beverley De-Gale, co-founder of African Caribbean Leukaemia Trust (ACLT), said Mrs Brown started a second round of chemotherapy at King's College Hospital in London last Friday in the hope the surgery would go ahead.
Mrs Brown, 23, who lives with her husband Mike and two-year old daughter Selina-May in Weymouth, had been told her only chance of survival was an urgent stem cell transplant.
Mrs Brown said she was "overjoyed" by the news of the visa u-turn.
"I would like to thank the British public and beyond, and my MP Richard Drax for their overwhelming support. I would also like to thank ACLT," she said.
"I will forever be grateful for the love and support they have shown my family and me."
Medical tests identified Martha as a perfect match, the ACLT said, but she was initially refused a visa because her teacher's salary of £222 per month was too low.
The charity, which set up the petition, said Mrs Brown had offered to cover all of her sister's costs.
A woman undergoing treatment for leukaemia said she was "overjoyed" that her sister has been granted permission to travel from Nigeria to donate bone marrow.