Qatar's Al Attiyah, driving a Toyota, finished second behind French 13 times winner Stephane Peterhansel in the 410km ninth special stage from Wadi Al Dawasir to Haradh in the Saudi Arabian desert.
Two-times winner Sainz, who had led by more than six minutes at the overnight bivouac, was fifth in his Mini buggy.
"In the first part we lost a little bit of time, we lost the way and we lost five minutes or something like that," said the Spaniard.
"After, I don't know...the tread came out of the tyre and we needed to stop. No puncture, it's like delamination of the tyre. It was not a good day for us".
Peterhansel thrust himself back into the overall reckoning, now six and a half minutes off the lead after his third stage win of the 2020 event.
Double Formula One world champion Fernando Alonso had another strong day and moved up to 10th overall in his Toyota.
Three days remain in the endurance event, being held in the Middle East for the first time.
Chilean Pablo Quintanilla was the motorcycle stage winner after Monday's timed action was cancelled following the death on Sunday of Portuguese veteran Paulo Goncalves.
Quintanilla, second overall on a Husqvarna, trimmed American Honda rider Ricky Brabec's overall lead to 20 minutes and 53 seconds.
Toby Price, the defending champion for KTM, was second on the stage and third overall, 26 minutes off the lead.
The Australian was first on the scene of Goncalves' accident and said he finished Sunday's stage "dehydrated with tears" after a failed effort to save him.
"No doubt this is going to be one of the toughest stages we’re all going to have to tackle and not because of the terrain or navigation," the Australian posted on Instagram before Tuesday's start, with a heart emoji.
He had earlier explained how he came over a crest, saw a rider down ahead of him on the dunes and knew it was serious.
He was joined by Slovak rider Stefan Svitko and retired French Alpine skier Luc Alphand, who was on the first helicopter to land. "Helping holding drip bags, getting bags of medical equipment and guiding other riders around a bad scene. We all worked as long as we could but there was nothing we could do," continued Price.
"I helped assist carrying him to the helicopter as it was the right thing to do. I was first at his side and wanted to be the last to leave." (Reporting by Alan Baldwin in London Editing by Peter Graff and Pritha Sarkar)