On the brink of elimination, the Devils came out all guns blazing and built a 2-0 lead on goals from Steve Bernier and Ilya Kovalchuk.
However, the visiting Panthers clawed their way back to 2-2 before the end of the second period after Kris Versteeg and Sean Bergenheim beat Devils' netminder Martin Brodeur.
Despite squandering a two-goal lead, Zajac said New Jersey had gone into overtime confident of getting the winner.
"“We said, 'It's there for us, we just need to push'. We did that, and we were able to get one," Zajac said. "“We have to duplicate this game."
The Devils, who outshot the Panthers 42-16, will travel to Florida for the series finale on Thursday.
"The possession certainly leaned toward the Devils tonight, that was the difference in the hockey game," Florida coach Kevin Dineen said.
"Playing as much defense as we did, it tends to wear you down. This was (New Jersey's) best game in this series, they had a heck of a lot of desperation."
Goaltender Scott Clemmensen made 39 stops for the Panthers, who are looking for their first play-off series win since 1996.
Wily veteran Brodeur made just 14 saves as he extended his decorated career with the Devils for at least one more game.
Brodeur, the most successful goaltender in NHL history, will be a free agent following this season along with team mate Zach Parise, leaving much uncertainty for New Jersey.
Parise contributed an assist to the winning effort while Kovalchuk's pass set up the winning goal.
The Devils have lost in the first round in three consecutive seasons but are trying to turn the corner.