Klingberg beat Calvin Pickard with a slap shot on a breakaway just moments into overtime.
The winner came after Jason Demers scored a power-play goal to tie it with 8:40 remaining in regulation. Before that, Niemi turned aside Nathan MacKinnon’s penalty shot.
Jason Spezza and Ales Hemsky each had a goal for the Stars, who won in Denver for the first time since Feb. 4, 2013.
Nick Holden, Blake Comeau and Carl Soderberg scored for Colorado.
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