Defending champion Egan Bernal (Ineos) finished in the five-man leading group and sits in second place overall, 21 seconds behind Roglic.
Yates, who made no mystery of his limited ambitions on the Tour, ended the stage in 15th place, 54 seconds off the pace, and dropped down to eighth overall.
As the race heads into its first rest day, when all the riders will be submitted to a coronavirus test, Roglic sits in pole position but Colombian Bernal, who relishes the long climbs of the Alps coming in the third week, is also where he wanted to be.
After attacking on the final climb on Saturday, Pogacar was on the offensive again on the ascent of the brutal Col de Marie Blanque, a 7.7-kilometre climb at an average gradient of 8.6 per cent, but Roglic, Bernal and Spain’s Mikel Landa (Bahrain-McLaren) followed.
Roglic picked up a five-second bonus at the top of the pass, situated 18 kilometres from the finish, and the small group caught former under-23 world champion Hirschi just shy of the line.
Pogacar was the fastest man to the finish as he gained back more time back after losing 1 min 20sec in the crosswinds on Friday.
France’s Guillaume Martin (Cofidis), who started the day second overall, slid down to third, 28 seconds off the pace, after losing touch with the top guns on the ascent to the Col de Marie Blanque.
The day after Thibaut Pinot (Groupama-FDJ) lost all hope of becoming the first local winner of the Tour in 35 years, Romain Bardet (AG2R La Mondiale) held steady, ending the day in fourth place overall, 30 seconds behind Roglic.