Charlton, who made seven changes from the side that beat Sheffield Wednesday on Saturday, seemed to be cruising through when they led 2-0 at half-time.
But the underdogs, in a scintillating 15-minute spell in the second-half, scored three goals to turn the tie on its head and it looked as though they may pull off a shock result before the home side's class told with a last-minute winner to finally shatter brave Stockport's hopes of glory.
Charlton took the lead on 36 minutes with a great 25-yard volley from their Bulgarian international Svetoslav Todorov and increased their advantage with a twice-taken penalty.
Izale McLeod was brought down in the area and took the spot-kick himself. It was saved by Stockport goalkeeper Conrad Logan, but referee Andy Hall ordered the kick to be retaken because of encroachment and this time Charlton's Chinese international Zhi Zheng stepped up to fire the ball home.
Nobody could have expected Stockport's onslaught as first Adam Proudlock brought them back into the game and Anthony Elding then equalised, both goals being scored from close-range.
The small army of Stockport supporters then went mad as Dominic Blizzard fired them into the lead with 21 minutes to go.
But there was still drama to come in this thrilling Cup-tie after Todorov had hit the bar with another fine shot, Charlton's Lloyd Sam equalised direct from a free-kick after he himself had been fouled.
With extra time looming Charlton snatched victory when Sam's corner was powerfully headed into the net by defender Paddy McCarthy.