Colombia keeper Faryd Mondragon set the record for oldest player in 2014 in Brazil by playing at 43 years and three days, so El-Hadary beat the record by a significant amount.
Al Muwallad took the spot-kick, but his firm strike was brilliantly pushed onto the bar by the veteran.
Essam El-Hadary then preserved Egypt's lead, the 45-year-old saving a Fahad al-Mulwallad penalty awarded for handball in the 41st minute.
Just when a draw looked on the cards, Saudi Arabia finally found the breakthrough in the 95th minute when Abdullah Otayf met Mohammed Al-Breik's right-wing cross with a first-time layoff into the path of Al-Dawsari.
The Egyptian FA confirmed Salah's inclusion in the line-up minutes after a tweet was published with a list omitting the star striker's name.
The Egyptians were faced with one clear task in this match: to get their first win in FIFA World Cup history.
Saudi Arabia played their best match of the tournament, showing quality football virtually throughout the entire 90 minutes.
With the final kick of the game, Saudi Arabia snatched the winning goal, with Salem firing a low effort past El-Hadary from six-yards out.
Salah's goal ended an 84-year wait for Egypt to score from open play, but the forward refused to go wild with joy amid reports he has fallen out the Egyptian FA.
El Hadary was unable to repeat the feat at the end of the half, going the wrong way and having no chance as Alfaraj scored from the spot.
The Al-Hilal man was key to Saudi Arabia's victory.
After weathering Egypt's dominance in the early stages of the first half, Saudi Arabia visibly grew in confidence, keeping possession for long stretches.