Referee Fred Graham ruled that Vale skipper Sam Collins had pulled back Constantine as he headed for goal, prompting furious dissent which earned bookings for Collins, Tony Dinning and Craig James.
Constantine drove his first spot-kick past keeper Mark Goodlad's right hand and was celebrating with team-mates when Mr Graham called him back after encroachment into the area.
But Constantine coolly slammed the re-take in off the underside of the bar for his 10th goal of the season.
It was no more than Torquay deserved for a brave second half performance in a match they had to win to peg back relegation rivals Milton Keynes Dons.
But the Gulls owed their victory as much to goalkeeper Andy Marriott as to Constantine's penalty.
Marriott produced three brilliant saves, highlighted by a first half tip-over from Michael Cummins' header.
Billy Paynter also hit the post with a second half header as Vale tried to catch Torquay on the break.
But the home pressure finally told when Zema Abbey flicked the ball on, Collins tugged Constantine as he ran into the box and Mr Graham did the rest.
After 18 games without a clean sheet, Torquay have now registered two in quick succession to breathe new life into their relegation fight.
As well as Marriott, they had an outstanding midfielder in on-loan Darren Garner from Rotherham.