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:
1. Anthony Patterson - 7
His passing was calmer and more precise than some of the outfield players and he made one excellent stop late on to prevent it becoming a nervy finish. 7 Photo: Frank Reid
2. Trai Hume - 6
Started superbly but grew frustrated by some of the inconsistent refereeing on show. Played his part in a good win in the end. 6 Photo: Frank Reid
3. Jenson Seelt - 6
Played some good passes and very incisive switches of play, though was caught out by the tempo of the game and the Birmingham press at times. Much better in thes second half and will be better for his full league debut. 6 Photo: Frank Reid
4. Nectarios Triantis - 6
Played an important part in the second goal, applying the pressure on Aiwu that forced him to turn into his own goal. One or two dangerous defensive moments but came through his league debut OK. Will be a boost for Mowbray, who knows he can trust this pair when needed next. 6 Photo: Frank Reid