But they did level from the spot at the second attempt six-minutes into first-half stoppage time when Ali Gabr was adjudged to have pushed Al-Muwallad over inside the area.
But Salman Al Faraj scored a spot-kick for Saudi Arabia seconds before half-time, just five minutes after El Hadary had saved Al Muwallad's effort, and Al Dawsari fired in the victor with seconds left to secure the Saudis' first World Cup win since 1994.
Egypt also gave 45 year-old keeper Essam El-Hadary a chance to become the oldest ever player to appear at a World Cup.
Egypt crashed out of the World Cup after losing to Russian Federation 3-1 on Tuesday, with Salah returning to action for the first time in three weeks following a shoulder injury but unable to save his team.
Uruguay defeated hosts Russian Federation 3-0 to seal top spot in Group A at the ongoing 2018 FIFA World Cup at the Samara Arena in Samara on Monday.
This is the first time Egypt has qualified for the Cup since 1990, so it's especially thrilling for a player like El-Hadary, who made his professional debut in 1993 and first played for the national team in 1997 against Algeria.
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.
On current form, Salah is one of the best players in the world, and the World Cup will be a poorer place without the brilliant Egyptian in the knockout stages.
Up until then Saudi Arabia had been on top, but had failed to test El Hadary.
Five minutes before the break, Saudi Arabia were awarded a penalty for a hand-ball in the box.
Salah was named in the starting line-up as expected and thrilled an Egyptian-dominated crowd of 36,823 by opening the scoring in trademark fashion after 22 minutes.
The victor came just after the allotted four minutes of stoppage time had elapsed, Abdullah Otayf playing a neat pass to Al-Dawsari who brilliant finished first time into the opposite corner.
The Al-Hilal man was key to Saudi Arabia's victory.