Berra, 27, was 9-17-2 with a 2.95 goals-against average and .897 save percentage with the Flames this season, his first in the NHL.
He spent the past four seasons with EHC Biel in the Swiss National League A before signing a one-year contract with Calgary. He is scheduled to be an unrestricted free agent at the end of this season.
He played for Switzerland at the 2014 Sochi Olympics. Berra stopped 30 of 31 shots in a 1-0 loss to Sweden during group play in his one start during the tournament.
"With this acquisition we feel we have further solidified our goaltending situation, for the present and future," Avalanche vice president of hockey operations Joe Sakic said in a team release. "He has already shown he can be successful at the NHL level and we are excited about what he can bring to our team."
Berra was a fourth-round pick (No. 106) in the 2006 NHL Draft by the St. Louis Blues. Calgary acquired the rights to Berra as part of a trade for defenseman Jay Bouwmeester on April, 1 2013, and signed him to a one-year contract 16 days later.