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5 October 2014 Last updated at 00:23 BST
"We had no idea that one day we could be forbidden from playing music, because music is universal," says Aliou Toure, the group's lead singer. "It's like being forbidden to see the woman you love. Music for us is like a woman we love."
They were traumatised by having to leave behind loved ones, but they also had some luck.
British musician and impresario and former Blur frontman Damon Albarn came to Mali in a gesture of solidarity, and Songhoy Blues was one of the bands he took under his wing.
The desert band's international career has taken off and they have played at several European festivals.
They also feature on Maison des Jeunes, the album that Damon Albarn and his Africa Express team recorded in Bamako.
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When Islamist militants captured Timbuktu and made music-making a crime, the members of Songhoy Blues felt they had no choice but to leave their homes and head south to the Malian capital, Bamako.