Froome, who is bidding to become the first rider to do a double of the Tour de France and the Vuelta in 39 years, set the stage alight with a vicious pace on the final climb of the 158.5 km stage, chaperoned by Sky teammate Gianni Moscon, although he was eventually caught on the descent.
He takes a two-second lead in the general classification over David de la Cruz (Quick-Step Floors), Nicolas Roche (BMC) and Tejay Van Garderen (BMC), while Nibali (Bahrain-Merida), who won the Vuelta in 2010, is fifth, 10 seconds behind the four-time Tour de France champion.
"It's a great surprise but I think it's the result of a lot of hard work today from my team-mates as well," Froome said.
"They did a fantastic job on the final climb. Especially a guy like Gianni Moscon. It's his first Grand Tour with Team Sky. He was great today, as were the rest of the guys."
Froome got to wear the Vuelta's red jersey for only the second time, the first since he upstaged Bradley Wiggins with a shock win in the individual time trial in 2011 and took the cycling world by storm in coming second in the general classification.
Froome recalled his 2011 performance when explaining why he had pushed so hard in stage three.
"I've lost a Vuelta before by 13 seconds so I'm going to fight for every second I can at this point," he said.
"It's been a long time (without wearing the red jersey). It feels amazing to put it back on and to be in this position. It's something I've thought about for a long time and obviously, I worked hard to be here after the Tour, so I'm really happy to be in this position."
Axel Domont (AG2R) took a 25-second lead over the escapees and a 4:40 lead over the peloton into the final 50km but his advantage rapidly eroded in the second climb.
Sky surged ahead in the final climb, led by Moscon with Froome on his wheel, with fellow general classification contenders Fabio Aru (Astana), Esteban Chaves (Orica-Scott) and Nibali close behind although Alberto Contador (Trek-Segafredo) was unable to sustain the pace.
Contador trails Froome by 3min 10sec, all but ending his chance of an overall win in his final professional race before retirement.
Froome led at the top of the climb with Chaves on his wheel but the pair were soon caught by Romain Bardet (AG2R) and Aru on the descent, with Nibali and Roche joining them in the final 700 metres
Roche made a futile attempt at breakaway as Froome slowed, but Nibali timed his attack to perfection to take the stage and 10 bonus seconds. De La Cruz finished second, taking six bonus seconds along with Froome.
“We're only on the third day,” Nibali said. “It wasn't easy to be ready for climbs and to win such a stage contesting the sprint.
“I had some difficulties on the last ascent, where my rivals keep a high tempo. I was behind and I must recover on the downhill to close the gap to the front small group of riders and try my chance to take the win
“I knew that it was difficult to win because there were very fast riders and today’s stage was really demanding due to the high temperature too. It's a long way to go.”
The Vuelta moves into Spain for Stage 4, a 198.2km route from Escaldes-Engordany in Andorra to Tarragona, Catalonia.
SBS will broadcast and live stream all stages of the Vuelta a España live from 19 August to 10 September.