Whereas Saturday's first leg was a cagey affair, this was anything but, as an assured and unchanged Stockport side controlled the return leg from the outset.
Barry, in his third loan spell for the Hatters from Aston Villa, was the difference as the hosts dominated the opening stages.
Having leapt out of the blocks, Barry almost sent Stockport ahead inside the opening five minutes when Stevenage failed to pick him up following a corner.
His effort was well blocked by Dan Phillips, but he made the difference soon after when he ran into the box, turned on to his right foot and coolly finished past Stevenage keeper Filip Marschall.
If Barry's finish was good, then Wootton's was even better.
Odin Bailey's whipped-in cross was well met by Wootton, who turned and volleyed the ball towards the top corner, with his effort deflecting off the head of Carl Piergianni.
Stevenage showed more fight after the break and could have laid the groundwork for a comeback had Goode's effort not been denied by the reflexes of Addai.