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The 36-year-old Nagelsmann, the second youngest national team coach in their history, signed a contract until the end of July 2024, the DFB said.
Rangnick steps in to replace Solskjaer, who was removed after United's 4-1 defeat by Watford this month.
Nagelsmann's contract at Leipzig ran to 2023 and Bayern reportedly agreed to a payment that could exceed 20 million euros ($24.13 million) for his release, making his one of the most expensive club coach signings in Europe.
Pierre-Emerick Aubameyang's 87th-minute header secured Arsenal a 3-2 win over Benfica for a 4-3 aggregate victory after the Premier League side had been staring at the exit door.
Tottenham Hotspur, Antwerp, Lille, Villarreal, Braga, Dinamo Zagreb, Red Star Belgrade, PSV Eindhoven, Granada, Bayer Leverkusen, Slavia Prague, Rangers and Benfica joined Milan in booking their places in the last 32 on the night.
Maradona's former club Napoli paid special tribute to the Argentine before a 2-0 win over Rijeka.
Bayern had not lost since December 7, 2019, and had gone 21 consecutive Bundesliga matches without defeat, winning their last 15.
The Bundesliga is the first major soccer league to return to action since the virus earlier this year shut down sports, and its return in a country that still has hundreds of positive virus cases a day is being closely watched by other leagues.
The result left City within a whisker of the last 16, as Group F leaders with nine points, three ahead of second-placed Olympique Lyonnais who drew 2-2 with Hoffenheim.
The Premier League champions were well below their best throughout and now face a serious challenge to qualify from a group which also features Germany's Hoffenheim and Ukraine's Shakhtar Donetsk.