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:
9. Jobe Bellingham - 7
Took his goal well and would have been disappointed not to get another for a similar chance at a corner shortly after. Excellent work rate throughout in what was at times a tricky game for Sunderland. 7 Photo: Frank Reid
10. Jack Clarke - 8
Another Saturday, another goal contribution. Persistent in driving into space and that eventually paid off when he teed Aouchiche up with an excellent assist. 8 Photo: Frank Reid
11. Nazariy Rusyn - 7
Really unlucky not to get his first goal with an excellent early effort that cracked the post. The ovation he received when replaced on the hour spoke volumes - his work rate was top draw and he caused real issues for Birmingham throughout. Finding his feet - impressive. 7 Photo: Frank Reid
12. Abdoullah Ba (sub) - 6
Caught in possession once or twice but made some dangerous runs, too. 6 Photo: Frank Reid