The first known escape was made by Bartemius Crouch Jr. Barty’s mother, already dying at the time, persuaded her husband, Bartemius Crouch Sr, to use Polyjuice Potion to switch her for their son, in case any of the prisoners were watching through their cells. The Dementors, being blind, could not tell the difference of when a healthy person and a dying person (Barty Sr. and his wife) entering the fortress, and when a healthy person and a dying person (Barty Sr. and his son) is leaving. Mrs. Crouch, disguised as Barty, died a few days later, in Azkaban.
This was the first known escape in the history of Azkaban, though it was not known to the public until thirteen years later.
The first (and only) unaided escape, however, was perpretrated by Sirius Black. This was officially the first one in Azkaban history (until Barty Crouch Jr.'s escape was revealed).