The 31-year-old left-back, who left Elland Road earlier this summer, has agreed a one-year deal with the option of a further 12 months.
City boss Steven Pressley told the club website: "We're delighted to be able to welcome someone of Danny's experience and quality. I've maintained that we only bring in players that improve the quality of the squad and Danny certainly ticks that box for us.
"He's shown throughout his career that he is not only defensively resolute but also poses an attacking threat which made him a definitive target for us as we build for next season.
"Danny will also be a positive influence on our younger players who continue to show promise and they will only benefit from his presence on the pitch and in the dressing room."
Pugh is the Sky Blues' sixth signing of the summer, joining Reda Johnson, Jim O'Brien, Danny Swanson, Ryan Allsop and Mohamed Coulibaly at the club.