Blair has signed an 18-month professional contract at Turf Moor.
The former Liverpool youngster has impressed since linking up with the Clarets earlier this season.
And the 18-year-old now gets the chance to further his career in the Academy set-up at Burnley.
Huddersfield-born Blair joined Liverpool's Academy from the Terriers in 2016 and played for the Reds' under-18s before his release last year.
He made his Burnley debut in the Clarets' FA Youth Cup defeat to Plymouth in January and has scored three times in his last three games for Danny Cadamarteri's youth team, which tops the U18 Professional Development League.