Klinkhammer put the Coyotes in front for good early in the second period, during which Phoenix scored three times in the span of less than four minutes.
The second goal from Klinkhammer was the last of the game and came with 5:47 to play.
New York was trying for a second straight win after ending a 10-game losing streak against San Jose on Tuesday.
Mike Smith picked up the win despite giving up three goals on 24 shots. All six Phoenix goals came against Kevin Poulin, who made 30 saves.