Vale have been making strides away from the drop zone in recent weeks but Oldham's dip could cost them their chance of finishing in the play-off places.
The Latics controlled long spells but they carved out few clear chances either side of Constantine's winning goal one minute before half-time.
Vale held on comfortably in defence as George Pilkington and Mark McGregor, who made his debut after joining from Blackpool, had impressive games at centre-half.
Oldham were hit by injuries up front with Chris Porter and 11-goal top scorer Andy Liddle both forced to watch from the stand.
They started brightly but Luke Beckett was unable to give them the lead when he seized on a pass by Richard Butcher and blasted his shot against the post from 10 yards.
Oldham striker Chris Hall headed just wide and also let fly from 25 yards as he tried to mark a rare start with his second goal of the season.
But the visitors shut up shop and broke away just before the break to go in front.
Constantine was given time and space just outside the Oldham penalty area and hit a right-foot shot which deflected off a defender and beat the dive on goalkeeper Lee Grant.
Vale pushed forward for a short spell after the break and saw Nathan Lowndes head over the bar from a Craig James cross.
Oldham's best chance came in the 52nd minute as Terrell Forbes crossed from the right but Beckett aimed his close-range header straight at keeper Mark Goodlad.