In the trade, Pittsburgh gets Morrow and a third-round pick in 2013, while Dallas gets prospect defenseman Joe Morrow and a fifth-round pick in 2013.
Joe Morrow was the 23rd overall pick by the Penguins in 2011. He is not related to Brenden.
Brenden Morrow, 34, is slated for unrestricted free agency in July. He's played his entire NHL career with the Stars since breaking into the league in 1999-2000.
"Brenden Morrow has represented everything we could ever ask for in a Dallas Star over the past 14 years, giving his heart and soul to this franchise. On behalf of the entire Dallas Stars organization, we wish Brenden nothing but the best going forward and thank him sincerely for his years of service," said Dallas Stars general manager Joe Nieuwendyk in a release. "That being said, we had a great opportunity to acquire a young top-end defenseman in Joe Morrow and we are delighted to add him to our organization. We identified Joe as an outstanding prospect and a target for acquisition if ever available, and we feel he can solidify our blue line for the foreseeable future."
The Bruins had also shown interest in Morrow, a source told ESPN.com.
Morrow for years was one of the NHL's premier power forwards, putting up a career-high 33 goals just two years ago in 2010-11 when he also played for gold medal-winning Canada at the Vancouver Olympics.
But over the past two years he's been slowed down by injuries. He's got six goals and five assists in 23 games this season and while he's not quite the same player he was before, he remains a quality character player who will bring experience, skill and toughness to the Penguins' lineup.
The Penguins are hoping to pair Morrow with former Dallas teammate James Neal and reigning NHL MVP Evgeni Malkin on the team's second line. Malkin remains out with an upper body injury but could return later this week.
Morrow is scheduled to arrive in Pittsburgh on Monday afternoon and could play as early as Tuesday night when the Penguins host Montreal. Penguins general manager Ray Shero said Morrow's veteran savvy and physical presence around the net will be a welcome addition.
"He's a guy that goes to those dirty areas and as you know when you get in the playoffs we know that's an important aspect of what we're trying to do," Shero said.
The trade came at a significant price. Joe Morrow was considered one of Pittsburgh's top prospects and Shero called him a player who will be "in the NHL a long time."
Information from ESPN.com's Scott Burnside and The Associated Press was used in this report.