Thomas completed the podium 47 seconds adrift of Quintana, but crucially extended his lead over teammate Chris Froome in the general classification standings from 1:39 to 2:31.
Defending champion Chris Froome nearly crashed out of the Tour de France - after being knocked off his bike when an overzealous policeman mistook him for a fan on stage 17 Wednesday.
Thomas holds an advantage of a minute and 59 seconds over Dutchman Tom Dumoulin and Froome appears happy to play a supporting role as his Team Sky colleague closes in on sealing a first Grand Tour title this weekend.
"I imagine we'll be able to finish it off".
The Tour will finish in Paris on Sunday, where Froome is bidding to make history by winning it for the fifth time and double Olympic champion Thomas is attempting to win it for the very first time.
"As soon as you get carried away, it's when it goes downhill", the two-time Olympic track champion said.
After a nine-month investigation - during which Froome constantly protested his innocence - cycling's governing body the UCI, on the advice of experts from the World Anti-Doping Agency, dropped the case just before the Tour began earlier in July. His team shared a supportive message on Twitter, and so did Team Sky. Yet when he arrived back at his team bus and responded to questions in French, he seemed dazed by the stage, a demeanour that was perhaps partly attributable to his collision with a policeman as he rode down from the finish.
Police had to remove bales of hay blocking the road 30 kilometres into the 218-kilometre leg from Carcassonne to Bagneres De-Luchon.
"Thomas was stronger than I was and I have to deal with that", he said.
"I just sprayed some water in the eyes and water in the face", the four-time Tour victor said.
But he lost the front wheel on a corner 6.5km from the end, and Alaphilippe zipped by in the polka dot jersey to collect his second stage win of the Tour.
At a news conference during the race's second rest day on Monday, Brailsford took umbrage at the treatment inflicted on his riders, who have been verbally abused and spat at over the past two weeks.