Weiss found the net 23 seconds into the game, putting the Panthers ahead for good, and added another score during a two-man advantage 1:12 into the second period.
Marcel Goc netted the eventual game-winner late in the middle frame for the Panthers, who won their first playoff game in 15 years -- the longest stretch between post-season victories in NHL history.
Jose Theodore turned aside 23-of-25 shots to earn the win.
Martin Brodeur made 20 saves in the loss.
The series now shifts to New Jersey for Tuesday's Game 3.
Notable deaths of 2014 [PHOTOS]