The deal Monday is the second the Sharks have made in the past week. San Jose traded defenceman Douglas Murray to Pittsburgh last week for two draft picks.
Handzus had one goal and one assist in 28 games. His biggest contributions came in shootouts where he converted three of seven chances. He was benched the last six games as the Sharks tried to get more speed in the lineup.
The pick San Jose got is one they sent to Chicago during last year's draft.
The 36-year-old Handzus is eligible to be an unrestricted free agent this summer. He played eight games for Chicago in 2006-07 before being sidelined by a season-ending knee injury.