The Liverpool forward netted a decisive strike in the final minute of the match in Borg El Arab to earn the Pharaohs a vital 3-2 victory.
With the score tied, Salah completed a clever one-two to break through into the box before lifting a careful finish over the goalkeeper.
The result moved Egypt level on 12 points with Tunisia in qualifying Group J before their final fixture in the section, at Niger in March next year.
Georginio Wijnaldum was on target for the Netherlands, meanwhile, scoring the opening goal of their 2-0 win at home to France in the UEFA Nations League.