Pittsburgh also receives a third-round pick in 2013, while Dallas gets prospect defenceman Joe Morrow and a fifth-round pick in the 2013 draft.
Joe Morrow, no relations, was the 23rd overall pick by Pittsburgh in the 2011 draft.
Brenden Morrow is currently in the final year of his six-year contract worth an average of $4.1 million US per season, and he is slated for free agency in July.
"The way he plays the game is something we've wanted to try and add," said Penguins general manager Ray Shero said.
Morrow is scheduled to arrive in Pittsburgh on Monday afternoon and could play as early as Tuesday night when the Penguins host Montreal. 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 veteran winger registered six goals and five assists in 29 games this season with the Stars, with whom he spent his entire 12-year career. In 835 games with the club, Morrow has 243 goals and 285 assists.
Olympic gold in Vancouver
Morrow also won an Olympic gold medal with Canada at the 2010 Vancouver Winter Games, along with Pittsburgh's Sidney Crosby and Marc-Andre Fleury.
The trade came with a hefty 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."
"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 Stars general manager Joe Nieuwendyk.
"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."