Egypt laboured for much of the first half but Liverpool star Salah managed to break the deadlock two minutes before the break following a fine effort.
Nantes striker Mostafa Mohamed doubled Egypt's lead on 55 minutes and Salah completed the rout with a well-taken penalty 12 minutes later to take his tally to 49 international goals.
It was an efficient rather than an enterprising display from the Pharaohs as former Benfica boss Vitoria starts his reign in convincing fashion.
Egypt will also face Liberia in another friendly on Tuesday before meeting Belgium in a high-profile game in November, a few days before the World Cup gets underway.
Vitoria played with a 4-3-3 formation in his first match in charge, fielding a familiar line-up that featured only one debutant in Pharco FC midfielder Mahmoud Hamada.
Egypt struggled for openings in a dour first half despite their territorial dominance and could only create some half-chances until Salah ended the resistance of Niger.
The visitors also had some efforts on counter attacks, including a powerful, angled shot from Victorien Adebayor that went just over the bar.
Egypt opened the scoring when Wolfsburg winger Omar Marmoush found Salah with a low cross to leave a defender on the ground with a superb piece of individual skill and drill into the far bottom corner with his left foot.
The introduction of Zamalek's in-form midfielder Emam Ashour in the second half added impetus and he superbly set up Mohamed for a second goal ten minutes after the restart.
His upfield pass broke the offside trap and released Mohamed, who dinked the ball over the keeper and into the net.
Salah made sure of the win when his powerful penalty went past the despairing dive of the keeper after Karim Doudou handled a Marmoush cross in the area.
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