Moen, 32, had zero points in 10 games with the Habs this season. The rugged checking forward has another left on his deal next year, which pays him $1.85 million.
"Travis is a proven forward that does all the gritty things needed to win," Stars general manager Jim Nill said in a statement. "He brings leadership and experience to our forward group and will be an important part of our penalty kill."
Gonchar had to agree to waive his no-trade clause in order to allow the trade. According to a source, though, he has been looking for a change of scenery since the summer.
He had one assist in three games this season, missing the start of the season with a foot injury. He will be an unrestricted free agent July 1. He's earning $5 million this year although a source told ESPN.com that the Stars retained eight percent of that salary in the trade.
The 40-year-old defenseman has 798 points (219 goals, 579 assists) in 1,256 regular-season games since making his NHL debut in 1994-95.
He has also played for the Washington Capitals, Boston Bruins, Pittsburgh Penguins and Ottawa Senators. Gonchar also represented Russia at four Olympic Games and won a Stanley Cup with the Penguins in 2009.
The Associated Press contributed to this report.