Soderberg, 33, had a career-high 23 goals last season and added 26 assists. The Swede completed his fourth season in Colorado after starting his career with Boston.
Arizona has a need for scoring punch up the middle. Clayton Keller led the team last season with only 47 points.
"Carl is a versatile, 200-foot player who has a history of being productive and playing tough minutes," Coyotes general manager John Chayka said in a statement. "He will be a stabilizer in our lineup and gives our coaching staff lots of flexibility."
Soderberg has one year left on his contract with a $4.75 million cap hit.
Connauton, 29, played mostly on the Coyotes' third defensive pairing. He had a goal and seven assists in 50 games and was a minus-2. He is under contract next season with a $1.375 million cap hit.
The Avs took defenseman Bowen Byram with the fourth pick in the draft, and he could crack the lineup next season. But Colorado needed some veteran depth after offseason surgeries for both Ian Cole and Erik Johnson.
"I'd like to thank Carl for everything he has done for the Avalanche both on and off the ice," Avs general manager Joe Sakic said in a statement. "This trade allows some of our young centermen to have a chance to play a more prominent role on our team while adding an experienced defenseman to our roster."