The Black Cats fell a goal down in the 29th minute when George Hall converted a close-range chance for the visitors.
Sunderland responded, though, as Trai Hume equalised on the stroke of half-time, before Amad put his side ahead 15 minutes from the end.
The hosts were forced to hang on in the closing stages after a red card for Dennis Cirkin, but managed to claim all three points.
Here’s how Phil Smith rated the Sunderland players against Birmingham:
13. Adil Aouchiche (sub) - 7
Got a goal that has been coming, doing well to gamble and turn in Clarke’s cross. Must be close to a start now. 7 Photo: Frank Reid
14. Eliezer Mayenda (sub) - 7
Looked a real threat. One superb driving run and ferocious shot drew a top save from Ruddy. Looks a talent on this brief outing. 7 Photo: National World
15. Mason Burstow (sub) - 6
Looked dangerous and so unlucky that his late effort drifted just wide of the far post. Still no striker goal, but all looked dangerous here and did well. 6 Photo: Frank Reid