Roglic (Jumbo-Visma) posted a time of 47min 5sec to finish 25 seconds ahead of New Zealand's Patrick Bevin (CCC) with Rémi Cavagna (Deceuninck-Quick Step) third on 27.
He now leads Alejandro Valverde (Movistar) by 1min 52sec with Miguel Angel Lopez (Astana) third at 2min 11sec and Nairo Quintana (Movistar) at three minutes even.
“I try to go as fast as possible in every time-trial and that’s what I did today," Roglic said. "I’m happy with my performance. We’ll see in Madrid how it ends.
"I’ve won today and we’ll go day by day. It’s nice to have the leader's jersey. I hope this time we’ll keep it until the end in Madrid. It’s never enough of a gap. Had I won with 10 minutes, I wouldn’t be satisfied either."
The 36.2-kilometre course between Jurancon and Pau was full of winding country lanes and technically demanding descents that were always likely to challenge those more comfortable on the climbs.
Lennard Hofstede (Jumbo-Visma) was the first of the 166 riders to start but Pawel Bernas (CCC) set a first serious reference with a time of 50min 52sec. His teammate William Barta is the first to break 50 minutes wit a time of 49min 17sec.
Benjamin Thomas (Groupama-FDJ) is six seconds faster and holds on to the hot seat for about 10 minutes, only to be edged by Cavagna's time of 47min 32sec.
Bevin pushed over the climbs and powers to the best time in the final run in to Pau, two seconds ahead of Cavagna.
The general classification battle started with Tadej Pogacar (UAE Team Emirates) 11 seconds faster than Cavagna at the first intermediate point. But Primoz Roglic (Jumbo-Visma) comes in 21 seconds ahead of that benchmark and after 11.9km had made bigger gains.
Valverde drops 38sec, Lopez one minute three seconds and Quintana one minute 26 seconds.
Roglic kept pushing until the finish and caught Lopez in the last 500m to take both the stage win ahead of Bevin, and the red jersey ahead of Valverde.
"There’s a lot of La Vuelta left but I’m in a good position and I’m looking forward to the next days," Roglic said. "So far I’m feeling good, even though I already had quite some problems.
"I crashed on the first stage. It was not the best way to begin. The wounds from Andorra are still open. From the point of view of performance, I’m still doing well and looking forward to what comes next. I’m confident in the team. They showed already they’re strong.”
Stage 11 will see the peloton negotiate a 180km route between Saint-Palais and Urdax, live on SBS Viceland and streaming at the website and SBS OnDemand from 11pm AEST.
