Paul Gallacher had to be on his toes straight from the whistle and was tested after only six minutes when a half-cleared free-kick was thumped back towards goal by Danny Whitaker only for the goalkeeper to make a solid stop.
Norwich threatened for the first time in the 20th minute when Dickson Etuhu led Richard Walker a merry dance before striking a goalbound shot from 15 yards that Mark Goodlad was forced to parry.
Constantine should have added another goal to his tally when he was presented with a pinpoint cross from Jeff Smith in the 28th minute only for him to head the ball directly into Gallacher's waiting arms.
Etuhu once more reminded Vale Park of his presence when slack marking within the midfield almost cost the home side dear with the midfielder striking a powerful shot that Goodlad dealt with.
George Abbey looked lively on his return to Martin Foyle's starting line up and after having a long range effort go close earlier, he worried Norwich again in the 63rd minute with a dipping shot.
The Canaries' cage was rattled once more when panic in the box saw Gary Doherty rush his clearance against the body of Constantine, almost sending the ball into his own net if not for a Gallacher save.
Substitute Dion Dublin should have broken the deadlock in the 76th minute but, just like Constantine earlier, his free header fell harmlessly to the goalkeeper with the match heading to extra-time.
Dublin's introduction boosted the away side's attacking options and it almost paid dividends with the imposing striker having two chances go close in the first period of extra time.
Port Vale dug deep late into extra time with Whitaker receiving the ball on the edge of the box but his shot crashed back off the crossbar to keep the scores level.
After a frustrating half an hour and some terrible spot-kicks, the whole night boiled down to Smith's final penalty which he stroked home to take his side into the fourth round.