Team Sky star Froome attacked with 80 kilometres of the demanding 181km stage from Venaria Reale to Bardonecchia left and won the stage by two minutes and 59 seconds from Movistar’s Richard Carapaz.
Froome, who started the day in fourth place in the general classification, three minutes and 22 seconds off pink, now leads by 40 seconds from defending champion Tom Dumoulin, with just two days of the race remaining.
Froome’s fellow Briton Yates, who began with a 28-second advantage over Dumoulin, is now more than 35 minutes off the pace.
Froome told Eurosport: “I don’t think I’ve ever attacked with 80km to go like that before on my own, and got all the way to the finish.
“But the team did such a fantastic job to set that up for me. It was going to take something really special to try to first of all get rid of Simon, to get away from Dumoulin and [Domenico] Pozzovivo and to go from fourth to first.
“I wasn’t going to do that on the last climb alone so I had to try it from a long way back and Colle delle Finestre was the perfect place to do it – a gravel road which reminds me of the roads back in Africa,” added Froome, who was born and raised in Kenya.
“I tried to stay within my limits and stay within myself there, so hopefully we can finish this off tomorrow.”
The stage was considered the race’s toughest and contained more than 4,000 metres of climbing, traversing the highest point of the race at an altitude of 2,178m atop the Colle delle Finestre. Froome attacked shortly after the road turned to gravel halfway up the Finestre.
If Froome’s win was one of the most emphatic of his career there were reminders of the ongoing controversy generated by his adverse analytical finding for Salbutamol at the 2017 Vuelta a Espana.
At one point as he struck out alone, Froome, who has spoken openly about his need to control his asthma, was pursued by two spectators dressed as doctors and carrying a giant inhaler.
Froome is bidding to hold all three Grand Tour titles at the same time, following his fourth Tour de France crown and La Vuelta win last season.
Yates, meanwhile, endured a stage he will wish to forget. The Michelton-Scott rider began to struggle as soon as the riders hit the Finestre at the mid-point of the stage, at the same time as Froome began to turn on the style.
There is one more mountainous stage today, a 214km leg from Susa to Cervinia, before tomorrow’s mostly ceremonial finish in Rome.