Tension flared early and there were six yellow cards but the game ended without serious incident. Saturday's encounter between the sides was marred by violence with the Algeria team bus being stoned two days before the match.
The breakthrough came in the 40th minute when defender Antar Yahia's volley clipped the bar and ducked in after a long cross from Karim Ziani.
Algeria adopted a more defensive game after the break but Egypt dominated play in pursuit of an equaliser.
Egyptian playmaker Mohamed Aboutrika valiantly created opportunities for the Pharaohs but Algeria's defence held tight.
Striker Emad Motaeb came close to equalising in the middle of the second half but his close-range shot was blocked by goalkeeper Faouzi Chaouchi.
"After (what happened in) Egypt, this was an important victory. I want to thank the people of Algeria for supporting us to the end. I want to thank Sudan for welcoming us. I'm so happy, I'm so proud," Algeria forward Kamel Ghilas told Reuters.
Sudan imposed a security crackdown involving an extra 15,000 police for the match, fearing a repeat of the trouble around Saturday's game in Cairo.
Egypt's 2-0 win there forced the Khartoum playoff, with the teams ending equal on points in Group C.
Algeria last qualified for the finals in 1986.
