Substitute Javi Martinez headed in the winner in the 104th minute after Europa League winners Sevilla had taken a 13th-minute lead through Lucas Ocampos’s penalty spot kick.
Bayern Munich levelled through Leon Goretzka strike in the 34th minute.
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The Bavarians, unbeaten run now in 32 consecutive matches, they also won the domestic league and cup double as well as the Champions League after Flick took over last November 2019.
Their last defeat dates back to December 2019.
Bayern Munich were the better side throughout and missed a bagful of chances in the first half, Yet it was Bayern Munich goalkeeper Neuer who kept them in the game first with a one-on-one save against Youssef En-Nesyri to send the game to extra time.
He then pushed another shot from the Sevilla player onto the post two minutes into extra time.
The Spaniards gradually ran out of steam, however, and it was only a matter of time until Bayern Munich scored again.
Martinez, who is reportedly close to agreeing to a return to Athletic Bilbao this season, had come on five minutes earlier and benefited from being given too much space to head in on the rebound.
The Spaniard also scored in extra time in Bayern Munich’s only previous Super Cup win in 2013.
Some 15,000 fans were seated in Budapest’s Puskas Arena with the game seen as a test event for the gradual return of fans to European games.
The match was held amid a growing second wave of COVID-19, with Hungarian cases at record highs, which had prompted warnings of the possible consequences of mass gatherings such as this game.