Begin, 30, who had been a healthy scratch in the last five games, had expressed a wish to be dealt to another team if the Canadiens were not going to use him.
The Trois-Rivieres, Que., native had six goals and four assists in 42 games this season. It was his fifth season with the Canadiens since he was claimed off waivers from the Buffalo Sabres in 2003. The former Calgary Flame had his best NHL season with Montreal in 2005-06 with 11 goals and 12 assists.
"Steve is a gritty, honest player with a very strong work ethic," Stars general manager Les Jackson said in a statement. "He's good on the penalty kill and he is a competitor in every sense of the word."
The six-foot-two, 211-pound Janik had one assist in 13 games with Dallas this season. The 28-year-old previously played for Buffalo and Tampa Bay before signing with Dallas a free agent in 2006.
Canadiens general manager Bob Gainey said Janik will be assigned to AHL Hamilton, if he clears waivers by noon on Saturday.
It was Montreal's second trade for blue-line help ahead of the March 4 NHL trading deadline. Last week, they picked up veteran Mathieu Schneider and a draft pick from Atlanta for a pair of draft picks.