Sunderland went 2-0 down after just 15 minutes thanks to goals from Amine Adli and a wonderful strike from midfielder Tyler Adams.
Enzo Le Fee got one back from the penalty spot just after the half hour mark, before Bertrand Traore restored parity almost immediately after the break.
Brobbey, 23, scored the eventual winner in the 69th minute, one week after scoring a dramatic equaliser in the 2-2 draw with Arsenal.
Speaking to reporters after the game, Brobbey was full of praise for the home fans, saying: "We always keep fighting for each other until the very end and that's what we did today.
"It was even better (than the goal vs Arsenal) and it feels really good to score the winning goal so I'm happy.
"The fans are like the 12th man, they are really behind you so it gives you a lot of confidence to keep going."