With the eyes of the world on him, Kevin Jorgeson clung to what was essentially a knife’s edge on a hunk of granite in Yosemite Friday and shimmied past a barrier that was threatening to end the rock climb of his life.

Jorgeson, 30, of Santa Rosa, got past a tiny, razor-sharp handhold on smooth-as-slate Pitch 15 on the cliff face known as the Dawn Wall on his second try late Friday afternoon, prompting an enormous cheer from spectators more than a thousand feet below.

The elated climber finally advanced after a week of suffering on the difficult section halfway up the vertical southeast face of 3,000-foot-tall El Capitan.

The effort to free-climb the Dawn Wall, a seemingly blank easel of granite on the right side of the mammoth rock that glows with the first morning light, is considered by many to be the hardest multi-pitch big wall climb ever attempted. It has been the dream of Jorgeson’s partner, Tommy Caldwell, for the better part of a decade.

Caldwell, 36, of Colorado had climbed past the most difficult sections and was poised to make a push to the summit. But he went down to lend moral support when Jorgeson could not get past Pitch 15 despite numerous attempts. Jorgeson fell on the so-called “crux” — really just a jagged pair of granite flakes the width of a thin dinner plate — at least four times earlier this week, ripping the athletic tape and remaining skin off his fingers.

“He gets to the last 10 feet of the pitch and can’t get by the crux,” said Tom Evans, a former rock climber who has set up telescopes and cameras in El Cap Meadow to follow the saga, which he blogs about every night. “It utilizes tiny very sharp holds, and his fingertips are so raw and cut that he can’t hold on and falls off.”

The battle resumed Friday evening under the watchful eyes of the world and documented by video crews and photographers who are swinging around on fixed ropes. If Jorgeson hadn’t made it, Caldwell might have had to finish the climb on his own.

Caldwell took the day off, rappelled down to Pitch 15 and was there when Jorgeson finally did what seemed impossible, shouting in elation after the feat, according to observers.

The assault on the Dawn Wall, which began Dec. 27, requires the two men to use only their hands and feet, with ropes and anchoring devices only to catch them when they fall. They cannot quit until the 32 pitches, which can be as long as 150 feet, are completed in sequence, according to free-climbing protocol. The climbers have to go back to the belay, or anchor point, and do the entire pitch over when they fall, which is often. After climbing each day, they sleep in slings on hanging portaledges thousands of feet in the air.

The men are essentially doing pull-ups with their fingertips wedged in cracks the width of a credit card, using tiny indentations on glacier-polished granite as footholds. This type of climbing requires immense strength, including vise-grip hands, superhuman pain tolerance, balance and perfect technique.

Caldwell, whose left index finger was lopped off in an accident with a saw years ago, is not merely famous for his climbing ability. In August 2000, he was one of four climbers on the 2,500-foot Yellow Wall in Kyrgyzstan who were taken hostage by Islamic rebels, according to accounts by Greg Child in Outside magazine and later a book.

The four eventually escaped after Caldwell pushed one of their guards off a cliff, a dramatic ending that caused an international sensation.

His wife, Rebecca Caldwell, wrote Thursday that she had “tears welling in my eyes,” after learning her husband had reached a section of the Dawn Wall known as Wino Tower.

“Wino was the place that I felt if he made it that far ... this thing would be possible,” she wrote in a blog post. “I didn’t realize he felt the same way, too. He stood atop Wino Tower with tears in his own eyes.”

Peter Fimrite is a San Francisco Chronicle staff writer. E-mail: pfimrite@sfchronicle.com Twitter: @pfimrite