The 43-year-old racing in his home country completed the 967km special stage, spread over two days, in 10h 56min 54sec, despite a 2min penalty for speeding.
He was followed by South African Henk Lategan at 4min 16sec with Qatari Nasser Al Attiyah, who overtook Al Rahji briefly to hold the lead for 142k, finishing third.
"It was really, really hard. I feel like this is our tenth day on the Dakar," said Al Rajhi on his arrival at the bivouac of the rally, which began on Friday.
"The navigation was very, very difficult in some places, due to the divots and dust. You needed a rocket, not a car to pass through them. It wasn't easy."
For this long stage, which started on Sunday, the competitors had to bivouac in the desert and did not benefit from the assistance of their teams at the night stop.
Sainz seriously damaged his Ford Raptor when he rolled it on Sunday but managed to finish the stage although the Spaniard is now 26th in the standings.
Nine-time world rally champion Sebastien Loeb, who is still looking for his first Dakar victory, was half an hour behind at the camp on Sunday evening, after a fan problem caused his engine to overheat.
But the Frenchman had a better Monday, making up much of the deficit to finish seventh, 13min 10sec behind the leader.
South African Lategan tops the provisional standings, 4min 45sec ahead of Al Rajhi with Al Attiyah in third. Loeb is sixth, 18min 56sec off the lead.
"We looked after the car for the stage because we knew it was really, really long," said Lategan.
"If you don't look after the car, it won't look after you. It’s actually a big surprise to be first because we haven't been really focusing on it. But I'm happy with that."
Third win for Sanders
Australian Daniel Sanders continues to dominate on the bikes, his victory in their '48-hour Chrono' making it three in a row after he also won the prologue and first stage.
It is the first time any rider has taken the first three stages since Spaniard Joan Barreda in 2017 between Bolivia and Argentina.
"It wasn't too bad, pretty hard in the soft dunes, it was very tough for a lot of us," said Sanders.
"When opening, you didn’t know if it was going to be a soft dune or a hard dune. It was pretty tough. The dust kind of ruined it a lot. Everyone was bunched up fighting in the dust for the opening bonuses. It was a bit tough on that side."
Sanders took victory on his KTM in a time of 11hr 12min 13sec, 6min 45sec ahead of Frenchman Adrien van Beveren (Honda) with American Skyler Howes (Honda) in third.
Sanders, who is aiming to become the second Australian to win the Dakar on a bike after Toby Price, holds a 12min 36sec lead over Howes in the provisional overall standings.
Botswana's Ross Branch (Hero) lies third, 4sec behind the American.
