The Scottish Medicines Consortium (SMC) has also accepted three new medicines for different types of cancer.
The SMC has licensed benralizumab (Fasenra) for the treatment of severe eosinophilic asthma, a type of asthma characterised by an excess of inflammatory cells known as eosinophils, which is not well controlled with other treatments.
Olivia Fulton, 33, from Edinburgh, has debilitating severe asthma which led her to give up her job as a nurse but could now be eligible for benralizumab.
She said: "My severe asthma leaves me gasping for breath, exhausted and unable to even walk down the road. While I've had asthma all my life, it worsened as I got older.
"I had no choice but to take long-term oral steroids at a high dose, which has left me with terrible side effects including osteoporosis.
"I used to be sporty and had my dream job as a nurse but my asthma got so bad I had to give it up.
"This new drug could transform my life, allowing me to get back into work and regain my independence. "
Dr Samantha Walker, director of research and policy at Asthma UK, said: "This kind of biologic medication could be a life-changing treatment that could signal hope to thousands of people in Scotland who have an acute form of asthma called severe eosinophilic asthma.
"This debilitating form of asthma is resistant to regular treatments such as inhalers and steroids, meaning many people are left dealing with terrifying asthma symptoms such as gasping for breath, or repeated trips to A&E."
Asthma UK said the drug was approved in England in January this year.
The SMC has also accepted patisiran (Onpattro) for the treatment of hereditary transthyretin (hATTR) amyloidosis, a very rare progressive genetic disease in which abnormal proteins called amyloid build up in tissues around the body including the nerves and the heart.
Symptoms include loss of sensation, weakness and muscle wasting, while most patients experience bladder problems, diarrhoea, incontinence and vomiting.
Patisiran has the potential to halt or slow the progression of the disease, and in some cases may improve symptoms, leading to a better quality of life.
Professor Julian Gillmore, consultant at the National Amyloidosis Centre, said: "hATTR amyloidosis strips patients and their families of hope and dignity and this decision is therefore an important development for individuals with this progressive and fatal disease.
"In its main clinical study, patisiran was shown to stabilise or improve symptoms in the majority of patients.
"Having this novel 'gene silencing' RNAi treatment available on the NHS in Scotland should now offer the possibility of preserving quality of life for longer than might otherwise have been possible."
The SMC has also licensed brigatinib (Alunbrig) for the treatment of a rare, advanced form of non-small cell lung cancer in patients who have not responded to another cancer medicine called crizotinib while durvalumab (Imfinzi) has been accepted for the maintenance treatment of another form of advanced non-small cell lung cancer.
The SMC has also accepted nivolumab (Opdivo) in combination with another medicine, ipilimumab, for the treatment of advanced renal cell carcinoma, a form of kidney cancer.
SMC chairman Dr Alan MacDonald said: "Our committee members are pleased to be able to accept these five new medicines for use by NHS Scotland."