A fortuitous goal by Alan Quinn, a sublime free-kick from Michael Tonge and a Jack Lester strike secured victory after an upset looked a real prospect as the League One side more than matched their Coca-Cola Championship opponents for 45 minutes.
Stockport had taken a shock lead in the second minute when Luke Beckett found himself unchallenged on the right-hand side of the area.
Beckett, a target for Sheffield Wednesday, took the ball to the byline before pulling it back - where it hit Blades defender Jon Harley and bobbled past wrong-footed goalkeeper Paddy Kenny's despairing dive to his right and into the net.
United were rocked and failed to find any kind of form, with Beckett's combination of physical presence, pace and cunning causing problems.
The Blades must have heaved a sigh of relief when he left the field injured after 28 minutes.
However, County should have gone further ahead and only a last-ditch tackle by Morgan prevented Chris Williams going through on goal.
Then Morgan proved his worth to the Blades by scoring an unlikely 36th-minute leveller when he nodded in Ashley Ward's header back across goal following Nick Montgomery's cross.
Ward should have put them further ahead moments later but he put a shot yards wide when unmarked in front of goal - and the same player hit a snapshot after turning on the edge of the area which Cutler had to be alert to gather down to his right.
United started the second half with far more conviction and purpose and when Barry Hayles played in Ward in the six-yard box the striker put the ball just the wrong side of the post.
The two combined again and Ward looked to have carved an opening for himself when he ghosted past two County defenders on the edge of the area - but an outstretched foot prevented his shot hitting the target.
Jack Lester replaced Hayles for his first appearance of the season and his craft almost reaped dividends in the 70th minute when he twisted and turned in the box and hit a shot across goal which flew just past the upright with Cutler beaten.
With just two minutes to go Lester was seemingly felled when poised to shoot in the six-yard box and the linesman signalled for a penalty.
The referee, however, disagreed and booked Lester for diving - much to the fury of the sparse crowd of 5,399.
It heralded a period of intense Blades pressure but Quinn miscontrolled the ball when well-placed in front of goal and the tie headed into extra-time.
Williams should have put County into the lead when he raced on to a through-ball after 11 minutes of extra-time - but he hesitated when confronted by Kenny and his eventual shot was gathered at the second attempt by the Republic of Ireland international.
Cutler then tipped Phil Jagielka's goalbound header over the bar in the last action of the first period.
However, Quinn made amends for his earlier error when his 25-yard shot hit the post and ricocheted off Cutler into the net to give the Blades the lead after 108 minutes.
Tonge then put the tie out of Stockport's reach when he curled in a free-kick from the left of the area in the 112th minute.
And Lester got the goal he deserved when he pulled the ball down and slotted the ball home after 115 minutes to give the hosts a slightly flattering win.