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:
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