Armee won his first grand tour stage by 31 seconds with a late attack from the breakaway group, dropping Alexey Lutsenko (Astana) in the final 600 metres.
Fabio Aru (Astana) finished the strongest of the general classification group, nine minutes and 56 seconds behind Armee.
At one point he had more than a minute on Froome, but the Sky leader launched a late attack along with Alberto Contador (Trek-Segafredo) and Michael Woods (Cannondale-Drapac) to trim the gap to Aru to 12 seconds.
Nibali could not keep up with the group as they approached the summit finish in Santo Toribio de Liebana, and lost 21 seconds to Froome, a day after cutting his overall lead to 1min 16sec in the Los Machucos mountains.
"That feels great especially after a difficult day yesterday," Froome said. "It’s good to bounce back again, and as I said yesterday, the morale is still good and the team is still strong. We’re just looking forward to getting through these next couple days now."
Contador, meanwhile, closed in on a podium finish in his last professional race as he moved one minute and 17 seconds behind Wilco Kelderman (Sunweb) in the general classification.
Froome has worn the leader's red jersey since the end of the third stage and is on track to become the first rider since French great Bernard Hinault in 1978 to win the Vuelta and the Tour de France in the same season.
The Vuelta stays in the mountains on Friday before a short, yet gruelling 117.5 km final mountain stage on Saturday which will determine the outcome of the race, before Sunday's final flat stage, which finishes in Madrid.
