The longtime hockey executive has been hired as president of hockey operations, a new position that puts Burke in charge of what President Ken King termed the "sports side of the Calgary Flames."
The Flames are in rebuilding mode after missing the playoffs for a fourth straight season.
Burke will report directly to King and general manager Jay Feaster will now report to Burke.
While it's not a normal arrangement in the NHL, it's common in other sports, Burke said at a news conference Thursday.
"This is a relatively new structure in professional hockey," he said. "There are two teams in the NHL that have this type of management structure. There are a number of teams in the National Football League that do, a number of teams in Major League Baseball that do and a number of teams in the NBA that do. And it works effectively. And it's going to work here."
Burke won the Stanley Cup as GM of the Anaheim Mighty Ducks in the 2006-2007 season. He was fired as president and general manager of the Toronto Maple Leafs in January. The 58-year-old Burke had been working as a part-time scout with the Anaheim Ducks since February.
He stressed that he will not be taking on the role of general manager.
"I'm not the general manager of the Calgary Flames, Jay Feaster is," Burke said. "Jay and I have been friends a long time, we met this morning, we talked about how to make this work, we're both committed to winning.
Burke likes the changes Feaster has made so far.
"I think I can provide some help, but I think Jay's got great ability," Burke said. "He's already made some important steps towards fixing things here and it's my pleasure and honor to be able to work with him."
King said the hiring of Burke in the works for many months.
"Both Jay and I feel very strongly that Brian will bring an added dimension of experience, guidance and mentorship that will add greatly to the progress that Jay has already made to improve our team," said King, adding Feaster has been involved in the process from the start.
Burke has had a lengthy career behind the scenes in the NHL.
He was director of hockey operations for the Vancouver Canucks for five years until the 1992-93 season when he became the GM of the Hartford Whalers. He stayed in Connecticut for a year before moving to the NHL's head office, becoming Commissioner Gary Bettman's executive vice-president and director of hockey operations.
In 1998, Burke returned to Vancouver to become the GM of the Canucks. He drafted several key players, including superstar twins Henrik and Daniel Sedin, as well as center Ryan Kesler. Team management elected to not renew Burke's contract after the 2003-04 season.
After spending time as a TV analyst, Burke returned to management, becoming the GM of the Anaheim Mighty Ducks in 2005-06 before winning his first Stanley Cup the next season.
He stepped down from the Ducks in November 2008 and two weeks later was hired by the Maple Leafs.
But the Leafs never made the postseason during Burke's tenure, with Toronto remaining the team with the longest playoff drought in the league. The Maple Leafs finally returned to the playoffs this year under new GM Dave Nonis, who replaced Burke.
"There was a lot that needed to be done in Toronto that took longer than I thought," Burke said.
But he believes the Flames are on the right path.
"I think Jay has already taken many steps to rebuild," said Burke, adding that he thought Flames had the best draft of any team in June.
A dual citizen of the United States and Canada, Burke's on the board of directors for Rugby Canada and is the director of player personnel for the U.S. men's hockey team at the 2014 Sochi Olympics.
He also was GM for the U.S. team at the 2010 Vancouver Olympics.