The Lightning had forged a 4-4 deadlock with 3 minutes remaining on Alex Killorn's power-play goal, but the Penguins then got their own man-advantage when the Bolts' Richard Panik was whistled for tripping with 57 seconds to go.
Penguins set up in the attacking zone and Evgeni Malkin sent a pass to Niskanen above the right circle. He one-timed it past Anders Lindback for the decisive goal.
Crosby struck for his eighth career hat trick and added an assist as Pittsburgh won for the fourth time in five games starting the season.
Marc-Andre Fleury stopped 14-of-18 shot for the Penguins.
Lindback finished with 21 saves for Tampa Bay.
Duggar sisters unveil Christian dating rules in new book
Beautician charged with giving client fatal silicone butt injection