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:
5. Niall Huggins - 8
Sunderland’s man of the match, in my opinion. A threat going forward and stood up to help a very inexperienced defence with some crucial interventions. Hugely impressive. 8 Photo: Frank Reid
6. Dan Neil - 6
Had a strong second half and was denied only by a good Ruddy save at one stage. Took time to get to grips with Birmingham’s pressing but came through. 6 Photo: Frank Reid
7. Pierre Ekwah - 5
Had a very difficult second half in which he often gifted up possession in the face of Birmingham’s aggressive press, and it could have left his side behind. Much better in the second half. 5 Photo: Frank Reid
8. Patrick Roberts - 7
Played a part in Sunderland’s second goal, his cross to the back post for Hume forcing the own goal. Another of the ‘second assists’ that Mowbray points to as a sign of his ongoing importance. Dangerous throughout and a relief that he was able to run off what initially looked like a muscle injury. 7 Photo: Frank Reid