Constantine's 85th minute strike cancelled out Ali Gibb's potential match-winner after Rovers haddominated throughout.
Rovers had the best of early action and came within a whisker of taking the lead with the game only three minutes old.
Craig Disley pounced on Vale's inability to clear a right-wing centre but shot narrowly over.
On paper the second round clash looked a mismatch as Vale started the game 32 places higher than the Pirates. But the travelling League Two side had the Valiants under seige for the majority of the opening period.
And they almost got their reward for 30 minutes of pressure when top scorer Richard Walker arrived unmarked in the box. But with the goal at his mercy the normally clinical striker amazingly steered the ball wide from point-blank range.
Despite Rovers' attacking endeavour and the Valiants' woeful backline, both sides went into the break level.
Despite a fruitless first-half assault Rovers weren't to be deterred and took the lead in the 53rd minute.
Mark Goodlad twice denied Walker in quick succession but the stopper had no answer when the ball fell at the feet of Gibb. The midfielder pounced on Walker's second saved shot and fired a low effort home from close range.
Prompted by his assistant, referee Scott Mathieson signalled a goal despite Goodlad's claims that he had clawed the ball off the line.
The goal was just the jump-start Vale needed and Michael Husbands almost levelled soon after.
Husbands latched onto a loose ball following a goalmouth scramble but saw his thunderbolt shot palmed wide.
With Vale pouring forward and chances coming thick and fast it wasn't long before they were back in the tie. Subsitute Mark Innes jinked his way down the right wing and looped a cross into the danger zone.
Husbands arrived right on cue and, with the goal begging, lashed in his third goal in Vale colours.
The home side had the momentum and pressed until the death for a winner. And Innes almost became the game's super sub in the 90th minute but snatched at a golden opportunity to win the tie.