Alejandro Valverde was the first to attack in the group of favourites in the intimidating ascent to La Camperona but he could not sustain the pace and lost time to overall leader and fellow Spaniard Alberto Contador as well as Britain's Froome.
Team Sky's Froome was the best of the top guns as he beat Contador by seven seconds and Valverde by 29, according to provisional results.
Contador retained the red jersey and now leads Valverde by 42 seconds and Froome, now third overall after Colombian Rigoberto Uran cracked in the finale, by 1:13.
Eleven breakaway riders reached the base of the final ascent, 8.3-km at an average gradient of 7.5 per cent and parts almost 20 per cent, with a five-minute advantage over the peloton.
Swiss Oliver Zaugg attacked in the steepest part and looked certain to win, only for Canadian Hesjedal, the 2012 Giro winner, to covercome him with 200 metres left and prevail after 200.8 kilometres from Santander.
In the group of favourites, Valverde sped away early and he was followed by Contador and Spain's Joaquim Rodriguez.
Froome appeared to struggle but it was the 2013 Tour de France champion who accelerated in the final kilometre to gain time over his rivals and send a warning.
Just as Contador seemed to be cruising towards a third Vuelta title, Froome's move showed that he will be a tough nut to crack and will be one of the favourites for Sunday's mountaintop finish in Lagos De Covadonga.
(Writing by Julien Pretot; Editing by Tony Goodson)
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