The 36-year-old Morrison had spent the past 1 1/2 seasons with the Flames and had totaled four goals and seven assists in 28 contests this season. In 923 games, with New Jersey, Vancouver, Dallas, Anaheim, Washington, and the Flames, he has collected 200 goals and 401 assists.
"We felt it was going to be difficult for us to find a spot for Brendan or get him significant ice-time," Flames general manager Jay Feaster said. "In Brian Connelly we obtain a 25-year-old highly skilled AHL All-Star defenseman who is still a legitimate prospect."
Morrison also tied a career-high earlier this season with a four-point effort on December 6 against Carolina.
Connelly, 25, has spent parts of the last four seasons as a member of the American Hockey League's Rockford IceHogs. During that time, he has compiled 21 goals and 105 assists in 211 regular-season games.