Pittsburgh still leads the series 2-1, with Game 4 in Ottawa on Wednesday.
Tyler Kennedy scored with just over a minute to play in the second period to give the Penguins a 1-0 lead.
It looked as though Pittsburgh had earned a 1-0 victory but Senators captain Daniel Alfredsson scored a short-handed goal with 29 seconds left in regulation time.
That set the stage for Greening's overtime heroics.
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