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.
Justin Bieber crashes Drake Bell's album release party
Rosie O'Donnell unveils nearly 50-pound weight loss