Yates, 24, came unglued when Dan Martin (Quick-Step Floors) attacked. The Irishman began the stage one second behind Yates in the overall classification, and by the day's end was 90 seconds clear into sixth.
"It was a very difficult day, I didn't have the best day and lost some time there in the final," Yates said. "I didn't really blow up mentally, I think I limited my losses OK."
The gap to the group ahead was only 20 seconds or so for Yates at the top of the Galibier, but a stiff headwind on the 25-kilometre descent to the finish, and only one rider for company, Dimension Data's Serge Pauwels, meant a difficult chase to the line.
"There was quite a lot of wind about and I got shelled off the group towards the end of the final climb. I really fought a lot to stay there as I knew there was still a long way to the finish but I couldn't do anymore."
Yates did enough to keep his rival in the best young riders' classification, Louis Meintjes (UAE Team Emirates), from threatening his white jersey. Yates finished 3:14 behind stage winner Primoz Roglic (LottoNL-Jumbo), but lost just 90 seconds to the South African, keeping a 2:28 advantage in the young rider competition.
"We are pretty far into the Tour now and I think I did really well to not lose too much time today," Yates said. "I went full gas on the final climb and I wasn't strong enough today so I hope to have better legs."