Reid had been confident of his plan to give Hearts more of the ball and try to catch them on the break, but patient build-up play from the visitors caused him to abandon his tactics in the second half.
But with extra-time looming Cole struck to break home hearts with a clean finish from the edge of the area after Turner's error in judging a long ball forward.
While it would further break Irish hearts to see a man they have adopted as one of their own swap Dublin for Abuja, Trapattoni may see this as possibly his last tilt at a World Cup.
Hibs still threatened on the break and Fletcher ought to have done better than slice a drive wide from the edge of the Hearts box after being found in space by a Stevenson pass.