Anaheim also acquired 20-year-old prospect Axel Andersson, a defenseman, in the deal completed three days before the NHL's trade deadline.
Kase, 24, drew interest leading up to the trade deadline as a cheaper option for those who couldn't land a big name like the New York Rangers' Chris Kreider. A productive two-way winger when healthy, Kase has another year left on his contract at a manageable $2.6 million, but health has been an issue.
He has missed the past five games with flu-like symptoms and won't join the Bruins until they return home after completing a four-game road swing at Vancouver on Saturday.
Kase scored 20 goals in his second season in 2017-18, but a variety of injuries limited Kase to 30 games last season, when he scored 11 goals and had 20 points. He has rebounded this season with seven goals and 16 assists in 49 games.
The Ducks are in rebuilding mode, sitting seventh in the Pacific Division after missing the playoffs last season for the first time in six years.
The Bruins benefit by trading the 35-year-old Backes, who has spent the past month playing in the minors. He is in the fourth year of a five-year, $30 million contract and had just one goal and two assists in 16 games with the Bruins before being demoted to AHL Providence last month.
The Bruins will retain 25% of Backes' salary.
Overall, the 14-year NHL veteran has 245 goals with 309 assists in 944 NHL games, spending the first 10 years of his career with the St. Louis Blues and the past four in Boston. A two-time U.S. Olympian, he was an All-Star in 2011 and was the runner-up in Selke Award voting the next season.
Andersson was playing for Moncton of the Quebec Major Junior Hockey League this season and had two goals and 20 assists.
The Associated Press contributed to this report.