The Rangers forced extra time with 6.6 seconds remaining in regulation after Washington's Joel Ward had been whistled for a high-sticking double-minor. Ryan Callahan redirected a Michael Del Zotto shot that Capitals goalie Braden Holtby saved, but Brad Richards put the rebound in off the crossbar for the equalizer.
Then, with New York still on a power play in overtime, Staal scored from the right point, handing the Rangers the advantage in their Eastern Conference semifinal series.
Anton Stralman scored and Henrik Lundqvist made 16 saves for New York, which will try to close out the series in Wednesday's Game 6 in Washington.
Holtby finished with 35 saves.
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