El-Hadary is two years and 158 days older than Colombian Faryd Mondragón, who previously held the record.
However, the Saudis were level just moments later when they were awarded a second spot-kick, after Ali Gabrin was adjudged to have pushed Al Muwallad in the box.
Al Dawsari's goal means that Saudi Arabia finish third in the pool on three points, while Egypt return home without a point to show for their efforts.
When Salah opened the scoring against Saudi Arabia, he chose not to celebrate.
Salah feels the Egyptian FA allowed him to be used as a political pawn and is now considering walking away from worldwide football.
The oldest player in World Cup history saving a penalty.
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Ahmed Hegazi elected to try and clear a cross when El Hadary could have caught it, and it fell for Hattan Babhir, who whipped in a right-footed volley that flew over the bar in the 54th minute.
He soon underlined his importance on the pitch though, calmly lifting the ball over advancing goalkeeper Yasser Al Mosailem in the 22nd minute after latching on to a raking pass forward.
Saudi Arabia face Egypt on Monday afternoon with both sides already out of the World Cup and now purely playing for pride.
A lengthy VAR review upheld the initial call, and El Hadary was beaten this time by Salman Al Faraj, making the score 1-1 going into halftime.
But Egypt will look to win its first-ever World Cup match, while the Saudis will try to avoid finishing last in Group play for the fourth time in four World Cup trips.
Both sides returned to the pitch for the remainder of the game after observing the 15 minutes compulsory rest.
Always available for the ball, the 28-year-old dictated the game for Pizzi's side with his penetrative passing and accurate crossing.