New York was protecting a 1-0 lead down the stretch and appeared ready to end the Devils' winning streak at two. In the space of 14 seconds, however, Clarkson and Zidlicky had scored to turn the game around.
Clarkson produced his goal with a wrist shot and Zidlicky took advantage of the power play that resulted when PA Parenteau was called for boarding.
Zidlicky's goal was his first of the season. The defenseman is in his first year with the Devils after having spent five seasons with Nashville and three with Minnesota.
Both the late goals came against Anders Nilsson, who made 33 saves. Johan Hedberg stopped 23-of-24 shots to get the victory.
The lone New York goal came from John Travers early in the third period.