The poet, who was diagnosed with cancer earlier this year, founded the library in Edinburgh in 1984.
She was also one of the chief movers behind its move to purpose-built premises in Crichton Close in 1999.
Scots Makar Liz Lochhead said: "Poets and lovers of poetry in Scotland will be very saddened to hear of the death of Tessa Ransford."
She added: "A fine poet of great sensitivity herself, she was the prime mover behind the establishing of the Scottish Poetry Library in Edinburgh - which dedicated library supports all kinds of poetry, serving poets, spoken-word artists, scholars of poetry, students, readers, and enthusiasts all over the world.
"Tessa was right. We need this and it will remain her lasting legacy, for which we are very grateful."
Ms Ransford was born in India in 1938 and moved back to Britain in 1944 and to Scotland at the age of 10.
She later attended the University of Edinburgh, where she studied German poetry.
After getting married in 1960 she moved to Pakistan where she undertook service as a missionary before returning to Scotland in the late 60s.
After publishing three books of poetry in the 1970s and early 80s, she turned her attention to setting up the poetry library.
In January 1984, the first library opened in Edinburgh's historic Tweeddale Court.
Ms Ransford steered the library through its first 15 years, overseeing its move to purpose-built premises in the Canongate before her retirement in 1999.
She was awarded the OBE for services to the Scottish Poetry Library in 2000.
The founder of the Scottish Poetry Library, Tessa Ransford, has died at the age of 77.