The Devils, who have not advanced past the first round of the playoffs in five years, got off to a good start behind their 19-year veteran goaltender.
New Jersey jumped to a 3-0 lead and then held on to claim home-ice advantage in a series that will continue Saturday with Florida hosting Game 2.
The goals by Zubrus and Carter came just 45 seconds apart late in the period.
Brodeur surrendered second-period goals to Sean Bergenheim and Kris Versteeg, but then slammed the door while improving his career playoff record to 100-82.
Theodore allowed the three goals on 38 shots.
The Devils and Panthers were the last of the playoff teams to get started in the opening round.
Notable deaths of 2014 [PHOTOS]
Michael Sam cut from the Dallas Cowboys