Stempniak, 32, joins his fourth team in two seasons and seventh of his NHL career.
The Buffalo, N.Y., native has nine goals and nine assists in 53 games this season, largely as a bottom-six player.
Stempniak provides more forward depth for a team in the first wild-card spot in the Western Conference.
He's on a one-year contract with a cap hit of US$900,000.
Klingberg, a 24-year-old 2009 pick of the then-Atlanta Thrashers, has spent most of this season with the AHL's St. John's IceCaps and has one goal in 12 NHL games.