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."
Barker took charge of Bury in April 2011 following Alan Knill's move to Scunthorpe and helped the club to promotion from League Two before guiding the Shakers to a 14th-place finish in League One last season.
A statement from Bury read: "The board of Bury FC would like to make clear that after having spent many hours over the weekend speaking to Richie in an attempt to convince him to stay, it became clear that the primary factor in Richie's decision to leave was his desire to be closer to his young family on the south coast."
Peter Shirtliff has been put in caretaker charge at Gigg Lane.