The 26-year-old inked a two-year deal with the Sky Blues to mark manager Steven Pressley's first summer signing, and the coach admits that Swanson is a "perfect fit" for his side.
"Danny is an exciting, technically gifted footballer who provides a real threat out wide and I'm delighted to get him in," he told the club's official website.
"We were not the only club looking to sign him so to get a deal done that brings him here is great news.
"He's got great footballing intelligence, he has a great attitude and he's hungry for success which makes him a perfect fit for us."
Swanson, who spent two years at London Road following a move from Dundee United, is believed to have snubbed a contract extension with Posh in favour of a switch to the West Midlands outfit.