The Black Cats took a first-half lead when Amad set up Ellis Simms to score in the 25th minute.
Birmingham saw more of the ball but fell further behind when Amad scored an excellent second for the Black Cats, converting in front of the travelling fans.
Substitute Lukas Jutkiewicz then pulled a goal back for Birmingham 12 minutes from time but Sunderland were able to hold on.
Our Sunderland writer Phil Smith has dished out his player ratings from St Andrew’s:
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