The Brit lost touch with the main contenders after Pogačar danced away 14kms from the finish on the Peyresourde, leaving Primož Roglič and Nairo Quintana to launch repeated attacks and apply the pressure to the favourites.
Yates rallied along with defending champion Egan Bernal (INEOS Grenadiers), Rigoberto Urán (EF), Mikel Landa (Bahrain), Guillaume Martin (Cofidis), Richie Porte (Trek-Segafredo) and Romain Bardet (AG2R) to reunite with the with the Slovenian and Quintana, all coming home together with Bardet a few seconds ahead after a small attack.
But Yates said the pressure was on earlier up the final climb as Jumbo-Visma turned the screws.
"Right at the bottom of the climb, Dumoulin set a ferocious pace and I just couldn't hold the wheel," Yates said.
"I had to ride at my own pace for a little bit sort of collect myself and then claw my way back and over the top.
"I think everyone was at the limit it was a really tough day," he said.
"It was a really tough day. In the beginning we let the break go. It was a good break for us I think the first guy was 17 minutes down or something so it was easy for us to control.
"Later on in the stage Jumbo-Visma come to the front and started pushing the pace really hard and then they rode pretty much full gas from when they got to the front until the finish."
"In the end I did my thing tried to stay with the best guys and here we are."
Mitchelton-Scott director Matt White said the team's 2020 Tour de France goals remain somewhat the same.
"Look this certainly wasn't planned," he said.
"But we're going to take it with two hands and run with it."
"The big picture - that doesn't change so while we're in the yellow jersey it's something we want to keep.
"It's the most prestigious jersey in our sport so we'll be running a similar plan for tomorrow trying to keep that jersey and then we'll have a rest day and a few flatter days before the next set of mountains.
"One day at a time but the plan coming in here still doesn't change. We want to win stages."
The Tour de France continues in the Pyrenees tomorrow, with another mountainous stage from Pau to Laurens set to test the peloton with steep climbs on the agenda. Watch the action from 8.05pm AEST on the SBS ŠKODA Tour Tracker and from 8.30pm on SBS HD and SBS On Demand.