A seesaw encounter at the Etihad Stadium saw Brobbey draw the Black Cats level twice in the first half, before completing his treble in the second period.
He took five shots in total, won 10/12 duels and drew five fouls to give City's defenders quite the headache and earn a 9.4 Flashscore player rating.
Brobbey also achieved a rare feat by becoming just the fifth different player in the Premier League to score a hat-trick and end up on the losing side.
The last player to do this was Roque Santa Cruz for Blackburn against Wigan in December 2007.
With clubs now heading into a three-week international break, Sunderland return from the pause at home to Brighton on Saturday, October 10th.