The Sky Team rider completed the 37-km course between the coastal towns of Javea and Calpe in 46 minutes and 33 seconds to win the 19th stage, his second victory in this year's event.
Colombian Quintana, wearing the race leader's red jersey, was the last man down the start ramp but could not maintain the pace of the Tour de France winner and finished 11th in the stage, losing two minutes and 16 seconds to Froome.
"Froome was flying," Quintana told Spanish TVE . "We still have an advantage and we need to defend it."
Spain's Jonathan Castroviejo finished 44 seconds behind Froome and Tobias Ludvigsson of Sweden was third.
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"I'm really happy with the result today," Froome said. "Time trials are painful but I get a lot of satisfaction when I'm told I finished first," he added.
"There is one more tough day of racing and let's see what happens."
Froome's hopes of winning a Tour de France and Vuelta double are still alive.
"I love the Vuelta," Froome said. "It's racing at its toughest and I've really enjoyed it."
Quintana is, however, still favourite to win his first Tour of Spain ahead of Saturday's mountain stage, a 193.2km ride from Benidorm to Alto Aitana before Sunday's sprint finish in Madrid.
"He has more than one minute but we are going to keep on fighting," Froome said. "We will see what happens tomorrow.
"Nairo has a good team around him. It's going to be tough to beat him."
Three-times Vuelta winner Alberto Contador finished eighth on Friday to leapfrog Esteban Chaves of Colombia into third in the general standings.
(Reporting by Adriana Garcia, editing by Ed Osmond)
