The loss marked the first time in this postseason that the Penguins have trailed in a series.
Pittsburgh now holds a 4-3 record in this postseason, and each of its three losses have come in overtime.
The Penguins outshot the Rangers 36-27 in the game, but were also tagged with 11 giveaways and were hapless defensively in front of Fleury for each of the Rangers' three goals.
Pouliot and Dan Girardi led the way with two points apiece.
Five-on-five drove play all evening -- both New York and Pittsburgh were 0-for-4 on power play chances. While New York hasn't scored a power play goal since Game 2 of their Quarterfinals series against the Flyers, Pittsburgh tends to rely on its ability to convert on special teams, making their inability to convert a big concern going forward.
Pittsburgh played a strong defensive game at times, settling down after a rough opening frame to hold the Rangers to just 12 shots and no goals in the second and third periods.
However, like the first two goals, a bad defensive-zone breakdown led to the Rangers' game winner.
Pittsburgh defenseman Rob Scuderi made a giveaway behind Fleury's net to allow the Rangers to set up shop, and a broken coverage in front of the net let Brassard get open in a soft area of the ice and pop one over Fleury's blocker.
Scuderi has had a rough season to date, and playing with postseason rookie and fellow not-so-good-at-skating defenseman Robert Bortuzzo means his immobility can be exposed, as it was on the Rangers' game-winner.
Though Scuderi was mostly good in this game, he and Bortuzzo were exploited for the deciding marker.
Offensively, Sidney Crosby was again held in check. Crosby drew immense coverage from the Blue Jackets in the first round, to the point that he and Evgeni Malkin had to be united in order to soften Columbus' defenders.
The two big centers were split up for Game 1, but if Crosby continues to draw such stifling defense, the move back to a line with Malkin could be queued up once again.
Game 2 is set for Sunday, in Pittsburgh.