Union twice hit the post in the 66th and 67th minutes in a very physical encounter, but it was their two goals in the first leg in Stuttgart that secured their spot in the top division next season.
The Berlin club, who will join Hertha as the capital's two representatives in the top division, had finished in the top 10 of the second division in each of the previous eight seasons.
"The team really deserved this after such a season," said club sports chief Oliver Ruhnert. "But it was the toughest route they chose. It is such a great feeling to give people what they have been yearning for."
Union, located in the eastern part of the city and known for their fierce fan loyalty and social engagement, become the 56th club to play in the Bundesliga.
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For Stuttgart, who had finished the campaign third from bottom, it was their second relegation in four seasons and their third in the club's history.
"Personally it is very tough to think straight at the moment," said interim Stuttgart coach Nico Willig. "At the end of the day Union are promoted and it is deserved. We failed to stay up. That was the mission and that is why I am very disappointed."
Hanover 96 and Nuremberg were relegated from the Bundesliga while Cologne and Paderborn won automatic promotion to the top flight.
(Reporting by Karolos Grohmann; Editing by Toby Davis)
