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.
"Odrej is a young, solid player, been a significant producer while 5-on-5, has shown versatility to be able to adapt his game and complement different lines," Bruins general manager Don Sweeney said.
The Bruins beat the Calgary Flames 4-3 on Friday night for their fifth straight win, opening a five-point lead on Tampa Bay atop the league's overall standings.
Kase 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.
He 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.
For the 35-year-old Backes, he no longer has to worry about whether he'll get another shot at playing in the NHL, being freed from what he described as "purgatory" with the trade.
Backes 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.
"It's been a roller coaster of, 'Am I just going to be dying a slow death here or is this something where I'm going to get another opportunity,'" Backes said during a conference call. "I was praying, actually praying, to get an opportunity where I could reassert myself and show that I still belong in the NHL."
The Bruins will retain 25% of Backes' salary.
Backes, who hasn't played since logging 8 1/2 minutes of ice time in a 5-4 win over Winnipeg on Jan. 9, expects it will take a week to get back into playing shape.
"I'm hoping to expedite that with that new energy I've got, the fire that I feel that's lit throughout my whole being," Backes said. "I'm coming at this with a full heart and ready to conquer and give everything I've got to each situation.
"I don't know what the last chapter will be. Hopefully, it's on my own terms and a glorious moment in Anaheim. But it may not be, and I'm ready for that, too. But I'm grateful to have another opportunity."
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.