The Red Devils turned to Barker to replace Sean O'Driscoll, who left the club last month to return to Nottingham Forest without taking charge of a single game.
A compensation package between the two npower League One rivals was finalised on Tuesday evening, with Barker moving to the West Sussex club on a two-year contract.
Barker told Crawley's official website: "It was certainly not a hard decision to come to Crawley, although it was difficult to leave Bury because I built up quite an affinity with the people there and they are a smashing club.
"But this is a fantastic opportunity for me. Crawley Town are a club on the up as their success in the last two years shows. There is only a week to go before the season starts so we have to hit the ground running but I am looking forward to working with the players and staff.
"League One is a big step up but Stevenage came through like Crawley did and reached the play-offs last season so it is possible for a promoted club to be successful. This is a great challenge for me in my managerial career and I am delighted to be here."
Source: PA
Source: PA