The Colombian (Movistar) held a six-second advantage ahead of the 36km ride from Jurancon to Pau, oddly entirely in France, but finished 3'06" adrift to leave his General Classification hopes hanging by a thread.
Quintana, who announced a move to lower-level team Arkea-Samsic on Monday, is 3’00” down on Roglic (Team Jumbo-Visma) and sits fourth overall.
Highlights: Quintana leaks time as Roglic wins
"It was quite bad. The feelings weren’t great, to be honest. It was very difficult. Primoz was good and we’ve lost a lot of time," he told reporters.
"Of course, I knew I was going to lose time, but I needed to try and find a way to recover. We’ve been in worse situations and we’re going to keep working hard to try and recover from this.
"Time trials are not really good for us, but we’ve been here before and we can still win."
Nairo Quintana: I can still win La Vuelta
Roglic leads Quintana’s Movistar teammate Alejandro Valverde by 1'52" ahead of Wednesday’s 180km Stage 11 to Dantxarinea but is refusing to get carried away following previous false dawns at Grand Tours.
The 29-year-old's tactics were questioned after he gifted his lead away at the Giro d'Italia in May, with Richard Carapaz (Movistar) eventually taking the title.
"It feels nice to have a leader’s jersey. I hope this time we keep it until the end in Madrid," the Slovenian said.
"It’s never too much of a gap. We didn’t have so much luck with the start, but I’m in a good position and I’m looking forward to the next days.
"I’m also confident in the team, they showed already they are strong. It’s just on us to do our best."