Orica-GreenEdge's Clarke outsprinted Germany's Tony Martin (Omega Pharma-Quick Step) on the summit of the second and final climb of the 160.5km stage to take a superb maiden win since turning professional.
But it was a nightmare day for overnight race leader Alejandro Valverde (Movistar), who was involved in a crash 30km from the finish and battled to limit his losses to his fellow race favourites.
The red jersey crossed the line 55 seconds slower than his rivals to concede the race lead to compatriot Rodriguez (Katusha), who finished in the main pack one minute and four seconds behind stage winner Clarke.
Clarke, a former national track champion, was part of an initial five-man break that formed inside the opening kilometres of the race and built up a maximum lead of 13 minutes in sweltering temperatures as high as 38 degrees Celsius.
"I've had so many second and thirds in my career and so I'm so happy to finally get my first win as a professional," said 26-year-old Clarke after using his track background to thwart Martin, the world time trial champion, in the deciding sprint.
"It was such a long day with so much wind at the finish," said Clarke. "I knew Tony would be strong and we worked together on the final climb. He's a very good time triallist but I knew I could take the sprint."
Rodriguez, second in May's Giro d'Italia, now leads Britain's Chris Froome (Team Sky) by one second on GC with fellow Spaniard Alberto Contador (Saxo Bank-Tinkoff) in third place, five seconds off the pace.
Twenty-four hours after pipping Rodriguez for the win in stage three, Valverde's luck turned on Tuesday after the 32-year-old former Vuelta champion was forced to ride out of his skin for the final 30km in a bid to defend his red jersey.
Despite a valiant effort, an isolated Valverde finished the stage 1:59 down on winner Clarke to drop to ninth place on GC, 36 seconds behind the new race leader Rodriguez.
The decisive moment of the stage occurred with around 30km left to ride just after the Sky team of Tour de France runner-up Froome had decided to increase the tempo on the front of the peloton, which trailed the leading group by more than 10 minutes.
Moments after the increase in tempo, a touch of wheels near the front of the pack brought down a flurry of riders - including the red jersey and several of his Movistar team-mates.
With their move made before the incident, Team Sky decided to press on and took advantage of the carnage created by not only the accident but also some savage crosswinds blowing across the plains of the Rioja region.
Three echelons in the peloton formed, with a fourth group - including Valverde - more than a minute off the pace as both Katusha and BMC contributed to drive the tempo on the front of the main pack.
As a result of the animation, the lead of the initial five man break came tumbling down. The gap was reduced to less than seven minutes at the start of the final climb, with Spaniard Luis Angel Mate (Cofidis) - who had won both intermediate sprints as well as crossed the summit of the previous Cat.1 climb in pole position - first to crack.
Assan Bazayev (Astana) and Jesus Resondo (Andalucia) also dropped off the front, leaving Clarke and Martin alone with 12km left to ride to the summit.
The main peloton soon reduced the arrears to four minutes, with the chasing Valverde group another 40 seconds off the pace.
Valverde looked to be making some headway on the steep early section of the climb but then Saxo Bank launched an attack through Contador and his trusty lieutenant Daniel Navarro. The pair was joined by Froome as well as Irishman Nicolas Roche, set to leave Ag2R-La Mondiale for Saxo Bank at the end of the season.
Froome and Contador soon dropped back to the Rodriguez group, and a new five-man chasing group formed around Roche. But Martin and Clarke held on to contest the win - as Valverde continued to struggle back to his main rivals.
With Clarke taking the win, it was the second time Martin had been denied victory in a mountain stage of a Grand Tour: in the 2009 Tour de France, the German was pipped by Spain's Juan Manuel Garate atop Mont Ventoux.
Kazahkstan's Bazayev held on for third place, 22 seconds behind the winner Clarke, while Spaniard Marcos Garcia of Caja Rural led the Roche chasing group over the line 55 seconds down, his arms raised in celebration after erroneously believing himself to be the victor.
It was a comical conclusion to an otherwise dramatic stage - and although Valverde will be fuming following his misfortune, the Spaniard will be aware that he is still only 36 seconds down on Rodriguez in the overall standings.
For many it was far worse: for a second successive day, Belgians Jurgen ven den Broeck (Lotto Belisol) and Thomas de Gendt (Vacansoleil-DCM) and Russian Denis Menchov (Katusha) had days to forget, with van den Broeck finishing 3:42 off the pace, de Gendt conceding 5:51 and Menchov, twice Vuelta champion, losing almost 10 minutes.
Following the best ride of his professional career, Orica-GreenEdge's Clarke moves to the top of both the points classification and the king of the mountains standings.
Wednesday's stage five of the Vuelta is a largely flat 168km circuit around Logrono which should re-open the doors to the race's sprinters.