Senegal also booked their ticket for the competition in January, beating Burundi 2-0 to join Algeria, Cameroon and Morocco in Gabon.
Egypt last played in the finals when they won the event for the seventh time in Angola in 2010 and have since missed out on three successive editions.
The Egyptians needed a draw in their Group G tie in Dar-es-Salaam and Salah put them ahead in the first half.
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Tanzania failed to convert a penalty in the second period before Salah scored again to seal victory.
Senegal went in front in Burundi when Sadio Mane netted from close range and a breakaway goal by Mame Biram Diouf had the Group K game in Bujumbura sewn up by halftime.
Guinea Bissau kept up their unlikely charge for a first finals appearance. They defeated 2012 champions Zambia 3-2 to extend their lead to four points in Group E but with nearest challengers Congo still to play on Sunday.
The former Portuguese colony are one of African football's long-standing minnows but now look set to reach the top 100 in the world rankings for the first time.
Mali won 3-0 in South Sudan with goals from Abdoulaye Diarra, Modibo Maiga and Moussa Doumbia and need a home draw in their last Group C game with Benin in September to reach the finals for a sixth successive time.
Uganda, who last qualified in 1978, kept up their hopes by beating Botswana 2-1 after Khalid Aucho scored the winner in Francistown.
The Cape Verde Islands continue to chase one of the two berths available to the best runners-up in the preliminaries after beating Sao Tome e Principe 2-1 with second-half goals from Ricardo Gomes and Nuno Rocha.
Sierra Leone's 1-0 home win over Sudan means holders Ivory Coast need a draw in their final Group I match to advance.
The Ivorians sank hosts Gabon 2-1 in Bouake on Saturday.
South Africa registered the biggest win of the day as Thamsanqa Gabuza and Keagan Dolly grabbed two goals apiece in a 4-0 triumph over Gambia in Banjul. However, it came too late to provide the 2010 World Cup hosts with any chance of going to the finals.
(Editing by Tony Jimenez)
