On the winning play, Adam McQuaid intercepted a clearing pass attempted by Dave Bolland at the Blackhawks' blue line and fed a pass to Tyler Seguin in the right circle.
Seguin then fired a cross-ice pass onto the stick of Paille, who unleashed a wrist shot that cleanly beat Chicago goalie Corey Crawford high to the glove side for the game-winner.
Patrick Sharp scored and Crawford finished with 26 stops for the Blackhawks, who will attempt to regain the advantage on the road in Monday's Game 3.
2014: NFL Cheerleaders [PHOTOS]
Michael Sam cut from the Dallas Cowboys