Briton Hugh Carthy, who surged into contention with an impressive win on the Angliru in the 12th stage on Sunday, was on-par with Roglic in the first part of the solo effort against the clock but the EF Education First rider conceded 25 seconds on the line.
Defending Vuelta a Espana champion Slovenia's Primoz Roglic claimed victory in a tough time-trial on Tuesday (Wednesday in Manila) to regain the lead in the overall classification with five stages remaining.
However, Carapaz couldn't overturn the 45 second gap, with Roglic leading the overall standings and taking the leader's red jersey by just 24 seconds.
"It's insane to think that I have won four stages", Roglic-already victorious on Stages 1, 8 and 10 on two uphill finishes and in a tough mass sprint-said.
"To take full control of the race, Roglič has to win by at least a minute", predicted Spanish TV commentator and former double Vuelta victor Pedro Delgado, but instead the Slovenian's advantage was a comparatively narrow 25 seconds on Hugh Carthy (EF Pro Cycling) and 49 seconds on Carapaz.
Carthy, meanwhile, has pledged to attack in the final hilly days, including an arduous penultimate stage on Saturday.
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The rider said that the stage "was one of the hardest races I've ever done".
Second-placed Carapaz still had energy in his legs to make a late dash towards the line, breaking away from Roglič as he crossed the finish 10 seconds ahead of his Slovenian rival.
Frenchman David Gaudu took the stage win for Groupama-FDJ from the breakaway, but all eyes were a further three minutes down the mountain with this the last chance to make a move in the general classification.
"Better be 39 seconds in the front now than 39 seconds behind".
"It's a super nice day, surprisingly I felt strong, I expected to suffer a lot more".
"Sometimes at the end you just need to stay under control and do you thing". Ireland's Dan Martin is further adrift, nearly two minutes off the front. This was the fourth round of tests in the Spanish race, which had been done before the start of the race in Irun, and on the first rest day.