Barkov skated in on Marc-Andre Fleury in the seventh round of the tiebreaker, moved left, and lifted in a one-handed shot to win it.
Florida's Roberto Luongo made a sprawling glove save, among his seven, on Chris Kunitz in the fourth round.
Jimmy Hayes scored twice in the third period to give the Panthers a 3-1 lead.
Nick Spaling deflected in a shot for a power-play goal with 4:52 remaining to bring the Penguins within 3-2, and Evgeni Malkin jammed a puck past Luongo to tie it with 3:10 left.
Patric Hornqvist had given Pittsburgh a 1-0 lead in the first period, and Jussi Jokinen tied it in the final minute of the second.
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