Uran attacked on the final climb of the day, the Cat.1 Altopiano del Montasio, to take a maiden Grand Tour victory ahead of Colombian compatriot Carlos Betancur (Ag2R-La Mondiale).
But what may have been a tactical ploy by Team Sky backfired when Wiggins was distanced by his rivals on the steep 20% ramps of the climb. The 2012 Tour de France winner finished the stage in 10th place and drops to 2:05 behind Vincenzo Nibali (Astana) on GC.
Italian maglia rosa Nibali outsprinted BMC's Cadel Evans to take third place on the stage and some vital bonus seconds that extend his lead to 41 seconds over the Australian veteran.
Uran's victory saw the 26-year-old Olympic road race silver medallist rise to third place on GC, one second ahead of team-mate Wiggins, whose form continues to flounder.
"I'm really happy to have won today," Uran told Eurosport. "It's a moment of great prestige for me to take such an important stage. Everything went perfectly for me and the team."
Uran's words will fan the flames of speculation that Sky have changed their tactics in favour of the Colombian as designated team leader Wiggins continues to struggle to rediscover the all-winning finesse of last season.
With the sun out and very little rain falling during Tuesday's stage, the 33-year-old could not even cite the adverse weather conditions as a factor - although there were still heaps of snow lining the roads of both decisive climbs.
Wiggins was not the only rider to suffer during the demanding 167km stage in north-east Italy, with defending champion Ryder Hesjedal cracking on the first of two Cat.1 climbs to see his chances of defending his Giro crown annihilated with a massive 21-minute loss.
Robert Gesink (Blanco) also suffered on the brutal slopes of the final climb, the Dutchman pedalling squares on his way to an exercise of damage limitation. Gesink recovered admirably to finish 1:16 down on the irrepressible Uran, but drops to fifth place on GC, 2:12 down on Nibali, who looked composed and strong over both climbs.
Betancur's second consecutive second place underlined Colombia's ascendancy in the world of cycling, the 23-year-old following in the wheel of Ag2R team-mate Domenico Pozzovivo to put in a late surge that saw him cross the line 20 seconds down on Uran.
Italian Mauro Santambrogio (Vini Fantini) took fourth place nestled between Nibali and Evans to rise to seventh on GC behind compatriot Michele Scarponi (Lampre-Merida), who lost touch with the race favourites on the final climb to finish a further two seconds behind Wiggins.
Polish youngster Rafal Majka (Saxo-Tinkoff) impressed in the final climb to take sixth place ahead of Pozzovivo in the Nibali chasing group. The 23-year-old joins compatriot Przemyslaw Niemiec (Lampre-Merida) to give Poland two riders in the top ten.
A duo from an initial break of 13 reached the foot of the final climb of the day with a small advantage over the peloton. But with Team Sky setting an fast pace in the main group, Venezuelan Jackson Rodriguez (Androni Giocattoli) and Dutchman Serge Pauwels (Omega Pharma-Quick Step) were caught inside the final 10km.
The main pack was already deprived of Canada's Hesjedal, who needed to be paced up the previous ascent - the Cat.1 Passo Cason di Lanza - by American team-mate Tom Danielson after imploding on the steep and winding climb.
Having suffered a jour sans on Sunday's stage to Florence, the Garmin-Sharp leader showed that he has not got the legs this year to make an impact. Hesjedal now lies in 33rd place on GC, a whopping 23:45 down on Nibali.
Sky started the final climb of Altopiano with Kanstantin Siutsou and Dario Cataldo pulling hard on the front ahead of Uran, while Wiggins dropped further back amid the main pack.
When Siutsou and Cataldo both peeled off with 8km remaining, Uran did not so much attack as simply increase his tempo as he rode off the front with apparent nonchalence.
Soon the gap was as large as 50 seconds with a select chasing group forming. But it became apparent that Uran's move was no platform for a counter-attack by Wiggins when the Briton failed to respond to Astana's determined chase.
One by one riders started to slip back, with Gesink, Wiggins, Scarponi and Spaniard Benat Intxausti (Movistar) feeling the heat on the hardest climb of the 96th edition of the Giro so far.
The diminutive Pozzivivo rode off in pursuit of Uran inside the final 3km and when the Italian was caught as the climb flattened out, he handed the baton over to his team-mate Betancur.
But Uran had done enough to secure the biggest win of his career - and in doing so give the Team Sky management a serious tactical headache ahead of the vital mountain stages to come. With just one second separating him and Wiggins on GC - and with Uran showing much more composure and bite in the hills - Wiggins may find his role in La Corsa Rosa reduced to supporting cameo.
Sky will not have to fit three pegs into one hole, however, after their second Colombian rider, Sergio Henao, struggled with the steep slopes to finish 3:14 behind his compatriot and drop out of the top 10, more than five minutes down on GC.
Nibali picked up vital bonus seconds in the second intermediate sprint 5km from the finish as well as over the finish line in third. Those 11 seconds saw the Sicilian increase his lead on Evans to 41 seconds on GC.
Evans's strong finish saw the 36 year old strengthen his hold on the red jersey points classification to 73 points to Elia Viviani's 60. Cannondale's Viviani was part of the day's main break and picked up maximum points at the first intermediate sprint in Paularo to rise above Britain's Mark Cavendish (OPQS) in the standings.
The Giro d'Italia continues on Wednesday with the 183km stage 11 from Tarvisio to Vajont - a mountainous but less demanding affair which should favour a breakaway.
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- Vincenzo Nibali
- Cadel Evans
- Team Sky