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Erik ten Hag's team looked on course to repeat their shock victory over City in the FA Cup final when Alejandro Garnacho put United ahead after 82 minutes, but City midfielder Bernardo Silva netted a minute from time to send the match to penalties.
English champions City started their big close-season signing, Norway's Erling Haaland but, in contrast to Nunez, he had a disappointing debut, missing several chances and struggling to combine with his new team mates.
The Sky Blues beat Chelsea 2-0 to become the first Premier League champions to win the trophy since Manchester United in 2013.
Gunners captain and match goal-scorer Pierre-Emerick Aubameyang blasted home the final kick of the afternoon to complete the north London side's second win over the league champions in the space of six weeks after a shortened summer break.
Pep Guardiola's side converted all five of their penalties in ruthless fashion while Georginio Wijnaldum had his effort saved by Claudio Bravo.
Results 1-5
