Terms of the multi-year deals were not released. Head coach Bob Hartley signed a multi-year contract extension in December, 2014.
The Flames improved 20 points in 2014-15 over the previous season and made the playoffs for the first time since 2009.
Hartley won this year's Jack Adams award as the NHL's top coach.
Calgary beat the Vancouver Canucks in the first round in six games and fell to the Anaheim Ducks in five games in the second round.
Cloutier, from Rouyn-Noranda, Que., is in his fourth season with the Flames and worked with Hartley for several years on different teams prior to their arrival in Calgary in 2012.
Gelinas, a former Flames player, has been an assistant coach with the team for three years. Pringle, from Picton, Ont., has been Calgary's video coach since 2011.