Constantine, signed permanently from Torquay following a successful loan spell, scored the opener from close range on 47 minutes after Chris Birchall's corner was flicked on by Sam Togwell.
The front man, who has now scored five goals this season, then timed a run from inside his home half perfectly on 65 minutes to latch on to Nathan Lowndes' through ball and rounded keeper Paul Rachubka to score with ease.
Victory was sealed on 78 minutes through Tony Dinning's penalty, driven low into the bottom left corner after Town defender David Mirfin was sent off for bringing down Lowndes in the box.
But the win should have been more emphatic in a match where the out-of-sorts home side, who have now lost twice and drawn three times since being paired with Chelsea in the FA Cup third round, hardly troubled the Valiants all night.
Vale should have gone in front on 16 minutes when Birchall's far post header from Constantine's by line cross was somehow clawed off the line by Rachubka.
Constantine also had chances either side of the break to add to his own and Vale's tally. He was put through by Dinning on 35 minutes but Rachubka saved well at his feet and then Lowndes fluffed the rebound.
Constantine then missed an easy chance on 75 minutes when Rachubka's scuffed clearance fell at his feet. The striker's attempted lob over the stranded Rachubka drifted wide of the target when it seemed easier to score.
Huddersfield, who had at least a dozen tame shots at goal that were either off target or straight at the keeper Mark Goodlad, went closest to scoring in the first half when Gary Taylor-Fletcher's flicked header from Chris Brandon's cross clipped the bar.