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:
Substitute Lukas Jutkiewicz then pulled a goal back for Birmingham 12 minutes from time but Sunderland were able to hold on.