Port Vale were coasting to their second win on the bounce when Andy Booth and Michael Collins struck to earn the Terriers their third win in a row and leave them undefeated in six.
But Town, now up into the play-off places, didn't make the brightest of starts on a blustery afternoon at a half-full Vale Park.
The visitors posed little threat in the first half and Vale immediately set about their task of building on an impressive victory on the road at Tranmere Rovers.
And the Valiants almost snatched an early lead when Jeff Smith drove wide with the game only four minutes old, even then it would have been no more than Town's sloppy defence deserved.
The home side needed another two efforts on Matt Glennon's goal, through Smith and Constantine respectively, before they eventually found their range.
Constantine blazed his first effort of the afternoon over before eventually finding the target on the 33rd minute.
The striker was gifted the simplest of tasks in nodding into a vacant net after Glennon had abandoned his station to flap at a Danny Whitaker's floated centre.
The hosts took a deserved lead into the interval but not before calling on Danny Sonner's 40th minute goal-line heroics.
Martin McIntosh's goal-bound drive from a Michael Collins corner-kick should have really ripped the Vale net off were it not for Sonner's last-ditch intervention.
Heartened by their late first-half flurry the Terriers looked a different prospect after the interval.
John McCombe's point-blank header forced acrobatics out of Mark Goodlad in the 55th minute, before he was eventually beaten twice in the space of two minutes.
Booth had only been on the pitch two minutes when he rose to power in Collins' far-post flag-kick before the home side hit self-destruct.
Booth's header had Vale reeling before Collins' hammer blow in the 78th minute laid them flat-out. The midfielder was allowed all the time in the world to pick out the top corner and with it win all three points.