Long-time leader Yates, who started the day 28 seconds ahead of defending champion Tom Dumoulin and three minutes 22 seconds ahead of Froome, ended the stage with his Giro hopes in tatters after blowing up.
Team Sky's Froome started the 185km ride from Venaria Reale to Bardonecchia in fourth place in the general classification but launched a devastating attack with 80km to go up the Colle delle Finestre to storm to a famous victory.
Yates, who began to struggle as soon as the riders hit the Finestre at the mid-point of the stage and cracked on the gruelling climb, finally crossing its gravel-road summit more than 15 minutes after Froome in 79th place. He now lies 18th overall.
“At the start, there was no signs of anything wrong. I was just tired, extremely exhausted and that’s bike riding, unfortunately. I tried to manage (the gap) but I really had nothing to give," Yates said.
“I gave everything today, it’s just how it is. But I’ll be back.
“It obviously hurts, but I have said all along that this is how I needed to ride this Giro to try to win it and if I achieved nothing more I can still be happy. And I will be.”
Australia's Rohan Dennis (BMC), who claimed the first Giro stage win of his career earlier in the week, crossed the line in 18th place on Friday and has slipped out of the top 10 to 13th in the overall classification.
“I expected today to be hard but I didn’t expect people to go all in on that climb [Colle delle Finestre]," Dennis said.
"I thought they would want to isolate a few people and test their legs out, and then have a real crack on the final climb. But, I was completely wrong and they just wanted to go all in and destroy the race, so it was a death march from there.
“At this stage, I think anything can happen but tomorrow (Stage 20) I really hope that everyone is feeling as bad as I am right now.”
Dimension Data's Ben O'Connor, did not finish the stage after he crashed on the descent of Sestriere. He injuries are being assessed by medical staff
The 22-year-old had ridden a surprising race in his first Grand Tour showcasing his pure climbing ability. He was 12th overall before the start of the stage and third in the young rider’s classification.