Al-Rajhi, driving a non-works Overdrive Racing Toyota, finished the 967km stage four minutes and 16 seconds clear of Lategan with Qatar's Nasser Al-Attiyah third for Dacia.
The Saudi was given a two-minute penalty after the finish of the two-day stage, and five-time Dakar winner Al-Attiyah four minutes, for exceeding speed limits in the desert.
That left Lategan and co-driver Brett Cummings, winners of Friday's prologue to the two-week event, four minutes and 45 seconds clear of Al-Rajhi.
"Towards the end of the really long dunes section, I got sick, had to stop and get out of the car for a bit," said Lategan, who took over at the top from American teammate Seth Quintero.
"After the dunes cleared, I started enjoying it a bit again.
"It’s actually a big surprise to be first because we haven’t been really focusing on it. But I’m happy with that. We’ve been playing a more strategic game over these two days."
Australian Toby Price and Britain's Sam Sunderland, both double winners in the motorcycle category and now teammates on four wheels, moved up to fourth overall with Overdrive Racing.
Nine-time world rally champion Sebastien Loeb staged an impressive recovery from lost time due to engine fan problems on Sunday and was in sixth place for Dacia and nearly 19 minutes off the lead.
Defending champion Carlos Sainz, who won with Audi last year and had hoped to be chasing a fifth title with a fifth different manufacturer, was an hour and a half behind Lategan after the 62-year-old flipped his Ford Raptor in the dunes on Sunday.
In the motorcycle category, Australian Daniel Sanders took his third consecutive stage win to take a 12 and a half minute lead over American Skyler Howes with Botswana's Ross Branch third.
Defending champion Ricky Brabec of the United States was in fifth place, 15 minutes behind Sanders.
Tuesday's 495km third stage is from the south-west town of Bisha northwards to Al-Henakiyah.