The 31-year-old, who has put pen to paper on a two-year deal, becomes Sky Blues boss Steven Pressley's second signing following the arrival of striker Mathieu Manset on Wednesday.
The club have been given special dispensation by the Football League to bring in both Manset and Webster.
Pressley told the club's official website: "We're delighted to bring a player of Andy's quality to the football club. "I played for a number of seasons alongside him so I'm very much aware of his footballing qualities and his character.
"I believe Andy will bring strength, leadership and aggression to our squad and will also be a vital stabilising influence on our young group of players.
"He is a player with good international and European experience but has served the vast majority of his career north of the border.
"After speaking with him it was evident to me he was ready for a new chapter in his career and I believe that this football club can provide him with a platform in which to do that."
Webster, who played alongside Pressley for five seasons at Hearts, also counts Rangers, Dundee United and Wigan as former clubs and has 28 full international caps for Scotland.