The triumph, in a gruelling endurance event now run entirely in Peru and featuring cars, bikes and trucks, was a first for Japanese constructor Toyota.
Spaniard Carlos Sainz, last year's car champion, won the closing 10th stage from Pisco with Al-Attiyah 12th and some nine minutes slower.
Sainz's compatriot Nani Roma, in an X-Raid Mini, finished second overall with nine times rally world champion Sebastien Loeb, runner-up in the stage, completing the podium with a privately-entered Peugeot in the absence of a works entry.
Al-Attiyah's previous Dakar wins were with Volkswagen in 2011 and Mini in 2015.
Overall runner-up in 2018, the 48-year-old led after every stage bar the second day in this year's event when South African team mate Giniel de Villiers was ahead.
"It’s fantastic. No mistakes during the entire rally. It was a very, very difficult rally for everybody," said Al-Attiyah.
"The good thing was that from day three we were leading and we built on our time every day until we came to the last day with a big, big lead.
"We felt when we came that in Peru...with no other country, there would be a lot of sand dunes so we could do what we know we're good at."
Price, battling a wrist injury suffered before the rally started, had led overnight but with only a minute between him and Chilean Pablo Quintanilla.
The Australian clinched his second Dakar title, on a KTM, after he won the stage -- his first victory of this year's event -- and Quintanilla fell.
"Pretty much all I can say is that it feels like there are about five people driving a knife in my wrist now," said Price after 10 agonising days of being jolted and bumped across the dunes and desert.
"Now I’ll just wait and see what damage I've done to my wrist. At the end of the day, the pain and torture has been worth it.
"I thought I would only be able to do two stages and then pull out and that would have been me done, but the support from everyone back home in Australia and then having some things go my way and a bit of luck, it just worked out in the end."
Russian Eduard Nikolaev secured his third successive Dakar title, and fourth in total, in the truck category with Kamaz.
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(Reporting by: Ossian Shine/Alan Baldwin in London,; Editing by Toby Davis)
