The Sharks announced the deal Monday. They acquired Burns from Minnesota during the draft in June for a package that included Devin Setoguchi.
Burns had one year left on his contract and general manager Doug Wilson said at the time he would like to lock up the 26-year-old long term.
Burns played 80 games last season, posting 17 goals and 29 assists. He also scored eight power-play goals and has the big shot the Sharks were seeking.
Burns has played 453 career games since joining the Wild as an 18-year-old first-round pick. He has 55 goals and 128 assists in his career.