Sunderland began strongly at City but were stunned in the ninth minute when Enzo Fernandez curled a free kick past goalkeeper Robin Roefs for his first goal since his record-equalling £125-million ($167.41-million) move from Chelsea.
The visitors deservedly levelled three minutes later when Brobbey took an Enzo Le Fee pass and placed a shot past Gianluigi Donnarumma but Rayan Cherki restored City's lead in the 29th minute when the ball fell to him in the box and he poked home.
Brobbey was a constant menace though and he equalised when he converted into an empty net from a Thomas Meunier assist.
City went in ahead after a breathless first half after Semenyo finished emphatically past Roefs.
Semenyo gave City breathing space in the 57th minute with a drilled finish after Sunderland's defence failed to clear the ball. But Sunderland, and Brobbey, were not finished.
The Dutch forward applied the finishing touch to a shot by Nilson Angulo to become the first Sunderland player to score a Premier League hat-trick since Jermaine Defoe in 2016.
He is also the first player to score a Premier League hat-trick and end up on the losing side since Roque Santa Cruz for Blackburn Rovers against Wigan Athletic 19 years ago.
Only some desperate defending stopped Sunderland levelling for the third time with Ruben Dias close to an embarrassing own goal before a late clearance. But Haaland finally got his goal to make the points safe, meaning he has now scored against all 25 Premier League clubs he has faced in a little over four years in England.
"It was very difficult, but we found a way to win the game. The good thing also is that these seven official games have all been different. We experience different things," Maresca said.
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