Intxausti - who wore the leader's pink jersey for one day during the opening week's individual time trial - gave his Movistar team their third stage win of the race after playing his cards right at the conclusion of the long 238km stage from the French ski resort of Valloire and back onto Italian soil.
The 27-year-old all-rounder waited for Lampre's Niemiec and Astana's Kangert to launch the final sprint before putting in his decisive surge to take the biggest win of his career and consolidate his position in the top ten on GC.
"It's a great day for the Movistar team. We have now won three stages, finished second in the team time trial and have worn the pink jersey," a delighted Intxausti told Eurosport after adding to previous Movistar wins by team-mates Alex Dowsett and Giovanni Visconti.
As he crossed the line, Intxausti made an X with his fingers in remembrance of his former team-mate and friend Xavi Tondo, who was killed in a training accident almost two years ago.
"It's May 21st today and May 23rd is a sad but special day for me because it's the second anniversary of what happened. I'm sure he would have celebrated my victory and so this win is for him, my grandfather and my team," Intxausti said.
Italian race leader Vincenzo Nibali (Astana) finished another flawless stage in a select group alongside his main rivals 13 seconds down on the leading trio to retain his maglia rosa with five stages remaining.
But stage 14 winner Mauro Santambrogio (Vini Fantini) cracked on the second of two climbs 20km from the finish to concede more than two minutes and drop from fourth place to sixth on GC.
Australian veteran Cadel Evans (BMC) still trails Nibali by 1:26 in second place, with Colombian Rigoberto Uran (Sky) in third at 2:46.
Twenty-two riders formed a strong break off the front of the peloton on the first climb of the day, the Cat.1 ascent of the Col du Mont Cenis - this time tackled from the reverse side - as the riders (minus BMC's Taylor Phinney, who was forced to retire with saddle sores) made their way back into Italian territory.
Boasting the likes of blue jersey Stefano Pirazzi (Bardiani Valvole), Italian veterans Danilo Di Luca (Vini Fantini) and Emanuele Sella (Androni Giocattoli), Dutch youngster Wilco Kelderman (Blanco) and stage 11 winner, the Lithuanian Ramunas Navardauskas (Garmin-Sharp), the break built up a maximum lead of five minutes.
They were kept in check by the Astana-led peloton, however, due to the presence of Italy's Damiano Caruso (Cannondale), who trailed Nibali on GC by just under 10 minutes.
Pirazzi inevitably took maximum points going over the summit of Mont Cenis to consolidate his lead at the top of the KOM standings ahead of a long descent towards the Susa valley.
The gap stood at four minutes as the leading 22 riders hit the long, flat 120km drag that characterised the stage's mid section. But pressure from RadioShack and Katusha - both without riders in the break - saw the lead tumble and created tension amid the escapees.
Only seven of the 22 fugitives remained on the front of the race for the second test of the day, the Cat.3 ascent of Andrate, 25km from the finish.
Danny Pate (Sky), Sella and Kelderman had made an initial move but were joined by Pirazzi, Navardauskas, Gorka Verdugo (Euskaltel), Grega Bole (Vacansoleil-DCM) and Jose Herrada (Movistar).
Team Colombia's Fabio Duarte broke clear of the main pack in pursuit of the leaders - and soon Duarte and Pirazzi were the only riders left out ahead of the streamlined group of race favourites.
Santambrogio suffered on the steep 13% mid section to drop off the pace as Nibali defended a series of attacks by Lampre's Michele Scarponi, with the likes of Evans and Uran sticking firmly to his wheel.
In the hunt for more KOM points to extend his lead in the blue jersey competition, Pirazzi put in a final dig close to the summit but he was rounded by Colombian Carlos Betancur, wearing the white jersey as the race's best youngster.
Ag2R-La Mondiale's Betancur crossed the summit 18km from the finish with an eight-second gap over the chasing group but was soon caught on the technical descent by Spaniard Samuel Sanchez (Euskaltel).
Nibali showcased his supreme descending skills on the sinuous ride back to the final flat run into the finish. But with all the main favourites shadowing each other's every move, Dutchman Robert Gesink spotted a chance to pounce and nipped clear alongside Poland's Niemiec, Estonian national champion Kangert and Intxausti.
Entering the cobbled streets of Ivrea, Gesink suffered a terribly unfortunate puncture inside the two kilometres ending his chance of glory.
The leading trio entered the final straight knowing that one of them would take the spoils. The pace slowed accordingly before Niemiec - the most experienced of the three - launched his sprint from far with Kangert hot on his heels.
Realising he had gone too early, Niemiec slowed and Kangert followed suit. But Intxausti simply supplemented his momentum with a powerful drive out of the saddle to secure his first victory on a Grand Tour.
Santambrogio crossed the line 2:09 down on the Nibali group to drop out of the top five. The Italian is now in sixth place, 4:57 down on Nibali. Lampre duo Scarponi and Niemiec rise to fourth and fifth at 3:53 and 4:13 respectively.
Britain's Mark Cavendish retains the red jersey but sees his advantage over Evans cut to just six points. The Omega Pharma-Quick Step sprinter will get a chance to build up a small cushion ahead of three back-to-back mountain stages in Wednesday's 214km stage 17.
Despite one Cat.4 rise towards the finish in Vicenza, the stage should conclude with the kind of bunch sprint that has seen Cavendish bring home four stage wins to date.
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- Vincenzo Nibali
- Cadel Evans
- Stefano Pirazzi