The Blues went behind in the first half after Isaac Olaofe's cross deflected off Jon Mellish and into the Carlisle net, but Paul Simpson's side forced extra time when substitute Omari Patrick netted late on in regular time.
After a thrilling two hours of football, the contest went all the way to penalty kicks, with 21-year-old Taylor Charters scoring the winning goal for his boyhood club.
Carlisle will return to the third tier of English football for the first time since 2014 after a memorable day at Wembley, while Stockport will have to settle for successive seasons in League Two.
The first half started in cagey fashion, with both sides understandably getting used to the magnitude of the occasion in the Wembley Stadium sunshine - Joe Garner was certainly relishing the challenge with a flurry of early tackles.
After 25 minutes, the game's first notable chance fell to the 35-year-old striker, but his headed effort via a free kick did not possess the required power to make it past Ben Hinchliffe in the Stockport net.
Olaofe was becoming an ever-increasing nuisance for the Carlisle backline, and the forward managed to get County into a first-half lead, with quite a bit of help from Mellish.
The 23-year-old picked up the ball on the right flank and his attempted cross deflected off the toe of Mellish, looping over the towering Tomas Holy to send the Stockport contingent at Wembley into delirium.
The Blues responded well to going behind and had a pair of chances to level the scores before the half-time whistle - the first arrived when Joel Senior burst through on goal, but a last-ditch tackle from Akil Wright prevented the right-back finding John-Kymani Gordon for a tap-in.
A few moments later, it was the turn of Gordon to race towards the Stockport net - this time Wright was beat by a quick change of direction - although the Crystal Palace loanee was never in control of his left-footed strike which flew over the bar.
The Hatters began the second half in promising fashion and had the chance to double their advantage in the game, but centre-back Wright was unable to guide his header on target despite having the freedom of the Carlisle box.
Carlisle made a host of attacking changes as they chased the equaliser, and in the 84th minute they were rewarded for being brave in their substitutes - cue crazy scenes between Hinchliffe's goal.
Stockport failed to clear the ball far enough after some neat play from the Blues from the right, and 27-year-old Patrick was quickest to the loose ball, with his first-time effort fizzing past the County keeper into the bottom-left corner.
Simpson's men understandably had the wind in their sails for the rest of regular time - the Blues' leading scorer Kristian Dennis had arrived off the bench and was showing glimpses of why he has netted 20 times in League Two this term.
Carlisle were inches away from avoiding the dreaded spot kicks when Dennis's glancing header was scooped off the line from County keeper Hinchliffe - it was a remarkable acrobatic save which kept the Hatters alive in the contest.
After failing to separate the two sides in extra time, the game went to kicks from the 12-yard spot, with Ryan Rydel missing Stockport's second and setting the tone for the shootout.
Both teams were flawless in their finishes after Rydel's miss, allowing Carlisle's academy graduate Charters to step forward and end his club's nine-season spell in the fourth tier of English football.No Data Analysis info