The Rangers rallied around St. Louis, who had been rocked by the sudden death of his mother before the game, and put in their best performance of the series.
"You don't expect these things," said St. Louis. "It was a couple tough days for my family, but I know deep down my mother would want me to play this game.
"She'd be proud of me coming here and helping as much as I can tonight. The boys are so supportive, the support I've got from the boys, my friends and family ... is unbelievable."
Pittsburgh lead the best-of-seven Eastern Conference semi-final series 3-2, with Game Six to be played in New York on Sunday.
Center Derick Brassard scored twice and the Rangers' much maligned power play also accounted for two goals. New York had been on a 0-for-36 power play skid.
The Rangers improved to 8-2 in their past 10 playoff games when facing elimination, including wins in three of five road games dating back to 2012.
Pittsburgh’s loss was its first in regulation in these playoffs. All three of its previous playoff defeats had come in overtime.
"We didn't get a win at home and it's a missed opportunity Game Five here at home," said Pittsburgh head coach Dan Bylsma. "We have to turn the page. We're going back to Madison Square Garden to win our fourth one there."
(Reporting by Andrew Both. Editing by Patrick Johnston)