Quintana, last year's Tour de France runner-up, pulled away in a controversial descent from the Stelvio pass and increased the gap in the final climb of the 139km trek from Ponte Di Legno.
There was confusion as organisers appeared to neutralise the race during that descent, made dangerous by the terrible weather conditions, but then told teams to ignore the instruction.
"Wrong communication: no neutralisation for the descent from the Passo dello Stelvio. Sorry for the wrong information," the Giro's official Twitter feed said in a message replacing an earlier one.
Not everyone was clear about it, however. "Neutralisation of the Stelvio-descent wasn't brought to all team cars. Big confusion," Team Saxo Tinkoff wrote on their Twitter feed.
Later on Tuesday, organisers said that they never intended to neutralise the descent.
"In consideration of audio recordings of instructions relayed to Directeurs sportifs during today's stage, the Directors of the Giro d'Italia would like to clarify that Race Radio provided an inaccurate interpretation of the indications stipulated by the Directors," they said in a statement.
"As previously stated, the intention was to guarantee rider safety during the first section of the descent (the first six hairpins, approximately 1500 metres) of the Passo dello Stelvio, where visibility was restricted due to low cloud and fog."
Riding his first Giro, Quintana finished the stage eight seconds ahead of Canada's Ryder Hesjedal and Frenchman Pierre Rolland took third place 1:13 off the pace.
Quintana now leads Uran, who finished the stage in ninth place and 4:11 down, by 1:41 with Australian Cadel Evans in third place overall 3:21 behind.
GRUELLING MOMENTS
"I don't understand why there is a polemic (dispute). I made up more time on the climb, we didn't make so much on the descent," said Quintana.
It was a day to remember as riders suffered gruelling moments in the Dolomites.
Rain froze on the TV cameras as the riders went through a snowy Gavia pass before taking on the feared Stelvio climb.
OPQS manager Patrick Lefevere used Twitter to direct his anger at the man responsible for the sport and technical area of the Giro, writing: "I don't know if this has something to do with modern cycling, Mauro Vegni. Giro d'Italia shame on you."
He then took aim at International Cycling Union (UCI) president Brian Cookson, asking: "Is this modern cycling, Brian Cookson?".
In the descent from the Stelvio, Spain's Dario Cataldo continued his efforts as confusion rippled through the peloton over the possible neutralisation.
Uran and Evans were dropped by a group featuring Quintana and Rolland.
Going into the final 22.4km climb, Quintana, Rolland, Cataldo and 2012 champion Hesjedal had a 1:30 lead over Uran and the other top contenders.
Cataldo was quickly dropped and, midway through the ascent, the lead had grown to 2:30 with Quintana doing all the work in the front group and accelerating on the steepest part 7.5 km from the finish.
Rolland and Hesjedal made it back on to Quintana's wheel but Rolland paid for his efforts and disappeared five km from the finish, while Hesjedal had to let go at the end.
(Writing by Julien Pretot; Editing by Alan Baldwin and Ken Ferris)
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