The 20-year-old Salford-born right-back has agreed a four-year deal at Turf Moor and arrives for Burnley's first season in the Premier League.
As a result of the player's age, the Clarets must now agree a compensation fee with United.
Eckersley made his first-team debut for the Reds in the 2-1 FA Cup victory over Tottenham back in January, coming on as a second-half substitute.