City took advantage of Arsenal’s shock 3-0 defeat at Brighton on Saturday but made life hard for themselves despite taking the lead three times at the Etihad Stadium.
Enzo Fernandez opened the scoring with his first goal since equalling the Premier League transfer record with a £125 million move from Chelsea.
The Argentine midfielder’s free-kick caught out Robin Roefs at his near post.
Sunderland won their first European match for 53 years against AZ Alkmaar in midweek but showed little sign of tiredness as City could not shake off the Black Cats.
Brobbey slotted home Enzo Le Fee’s pass for the first of his treble.
Quick feet and a slick finish from Rayan Cherki restored City’s advantage, but Brobbey again replied within four minutes at the end of a fine Sunderland move.
Antoine Semenyo smashed City back in front before half-time and thought he had settled the game with another accurate left-footed finish just before the hour mark.
Brobbey completed his hat-trick after Nilson Angulo beat Gianluigi Donnarumma to a ball over the top.
A couple of nervy moments from the Italian goalkeeper nearly led to a Sunderland equaliser as Ruben Dias was forced to clear off his own line.
However, Erling Haaland finally got involved in the goalfest to seal the three points and ensured he has scored against every side he has faced in the Premier League.
Victory extends Enzo Maresca’s perfect start to the daunting task of succeeding Pep Guardiola at City boss.
“To be honest, I prefer to win 1-0 than 5-3 but I expected a very difficult game,” said Maresca.
United head into a three-week international break just two points outside the relegation zone as they failed to ease the pressure on manager Michael Carrick in a 1-1 draw.
But it could have been even worse for the Red Devils after a calamitous Lisandro Martinez own goal put Fulham on course for their first league win of the season.
Senne Lammens parried Calvin Bassey’s strike back into the danger area and then could not clear when Martinez inadvertently turned the ball towards his own goal.
Cunha’s 89th minute equaliser also took a wicked deflection off David Affengruber to save United’s blushes.
The Red Devils have taken just five points from their opening five games and were dumped out of the League Cup by Brighton in midweek after surrendering a 2-0 lead at home for the first time in 50 years.
A point is enough to edge Fulham off the foot of the table as Tottenham drop to the bottom.
Liverpool won 1-0 on Andoni Iraola’s return to Bournemouth thanks to Alexander Isak’s winner.
Liverpool’s performance was far from convincing but they got the only goal when Cody Gakpo’s cross sat up for Isak to fire in his fourth of the season.
“To win here is very difficult, I know it very well,” said Iraola. “Overall we’ve improved, definitely since the first games.”
Victory moves Liverpool up to sixth and within a point of third-placed Brighton.
Leeds missed the chance to go third themselves after being held 0-0 at home by Crystal Palace. (AFP)
