Dougie Gair scored the only goal of the game from the penalty spot late on after Reece Donaldson had been sent off for conceding the spot-kick.
The 2-1 aggregate score also brought to an end East Stirlingshire's 61-year tenure in senior football.
Shire will now enter the Lowland League, which Edinburgh won this term.
The home side had the better of the opening 45 minutes with Max Wright and Tommy Orr both coming close to giving East Stirling the lead but City coped well, the experienced Joe Mbu at centre-half a rock at the back for the capital club.
In the second half City came much more into the game and Mark McConnell forced a fine save from goalkeeper Darren Dolan.
With four minutes left the tie took a dramatic turn when the visitors were awarded a penalty.
Ross Allum was through on goal when he was brought down by Donaldson, who had successfully appealed against a red card in the first leg.
Referee Don Robertson immediately pointed to the spot and issued the Shire defender a straight red card.
And Gair, who scored from the spot in last week's draw, hammered the ball down the middle of the goal to spark celebrations among the City fans.
And when the full-time whistle was blown, the players and fans rejoiced as they can now look forward to senior football next season.
Match ends, East Stirling 0, Edinburgh City 1.
Second Half ends, East Stirling 0, Edinburgh City 1.
Attempt saved. John MacDonald (Edinburgh City) right footed shot from outside the box is saved in the top centre of the goal.
Foul by Mark McConnell (Edinburgh City).
Neil McCabe (East Stirling) wins a free kick in the defensive half.
Corner,  East Stirling. Conceded by Sean Muhsin.
Foul by Ross Allum (Edinburgh City).
Lloyd Kinnaird (East Stirling) wins a free kick in the defensive half.
Goal!  East Stirling 0, Edinburgh City 1. Dougie Gair (Edinburgh City) converts the penalty with a left footed shot to the centre of the goal.
Penalty Edinburgh City. Ross Allum draws a foul in the penalty area.
Reece Donaldson (East Stirling) is shown the red card.
Penalty conceded by Reece Donaldson (East Stirling) after a foul in the penalty area.
Foul by Ian McFarland (Edinburgh City).
Max Wright (East Stirling) wins a free kick in the attacking half.
Attempt missed. Ross Allum (Edinburgh City) right footed shot from the centre of the box is close, but misses to the right.
Attempt missed. Mark McConnell (Edinburgh City) left footed shot from outside the box is high and wide to the left.
Corner,  Edinburgh City. Conceded by Neil McCabe.
Corner,  Edinburgh City. Conceded by Lloyd Kinnaird.
Substitution, Edinburgh City. Sean Muhsin replaces Ross Guthrie.
Chris McKee (Edinburgh City) wins a free kick in the defensive half.
Foul by Gavin McMillan (East Stirling).
Ian McFarland (Edinburgh City) wins a free kick in the attacking half.
Foul by Ross Fisher (East Stirling).
Shaun Harrison (Edinburgh City) wins a free kick in the attacking half.
Foul by Max Wright (East Stirling).
Substitution, East Stirling. Aron Lynas replaces Kristopher Faulds.
Corner,  Edinburgh City. Conceded by Darren Dolan.
Attempt saved. Mark McConnell (Edinburgh City) left footed shot from the centre of the box is saved in the bottom left corner.
Substitution, East Stirling. Gavin McMillan replaces Thomas Orr.
Attempt missed. Mark McConnell (Edinburgh City) left footed shot from outside the box is high and wide to the left.
Ross Guthrie (Edinburgh City) wins a free kick on the left wing.
Foul by Chris Townsley (East Stirling).
Attempt missed. Mark McConnell (Edinburgh City) right footed shot from outside the box misses to the left.
Attempt missed. Aaron Dunsmore (Edinburgh City) left footed shot from the right side of the box is close, but misses to the left.
Substitution, Edinburgh City. Shaun Harrison replaces Frazer Patterson.
Foul by Frazer Patterson (Edinburgh City).
Thomas Orr (East Stirling) wins a free kick in the attacking half.
Attempt missed. Ross Allum (Edinburgh City) header from the left side of the six yard box is just a bit too high.
Corner,  Edinburgh City. Conceded by Reece Donaldson.
Corner,  Edinburgh City. Conceded by Reece Donaldson.
Edinburgh City became the first club to be promoted to Scottish League Two through the pyramid system after a dramatic win over East Stirlingshire.