The 27-year-old turned down a new deal at Ashton Gate after his contract ran out following the club's relegation and undertook a medical at Burnley's Gawthorpe training headquarters.
Sean Dyche made his move quickly after the Turf Moor club were also left short on goalkeeping options following the departure of Lee Grant and long-serving Dane, Brian Jensen.
Meanwhile, former Sheffield United and Crystal Palace defender John Pemberton has expressed his delight at becoming Bristol City's new assistant head-coach as the club continue preparations for life in npower League One.
Pemberton, who made over 300 appearances for six different clubs during his playing career, will assist Sean O'Driscoll with all first-team duties while also overseeing the progression of the under-21 development squad.
The highly-respected 48-year-old had a lengthy spell coaching at Nottingham Forest before moving on to the Blades, who he guided the Blades' Academy to FA Youth Cup glory for the first time in 2011.
Following the departure of former coach Richard O'Kelly last week, the club moved quickly to appoint Pemberton.
"I'm delighted to be joining Bristol City Football Club and to be working alongside Sean again," the new coach told the club's official website.
"When Sean told me of the club's vision moving forward, I jumped at the chance to be able to assist with sculpting the future of both the club and its players."
The management duo formerly worked together at Forest, where Pemberton held the post of professional development coach.