DRESDEN, Germany ­— Running low on hope and almost out of time, the Americans were surely beat, about to make their earliest exit from the Women’s World Cup.

And then, with one of the most thrilling goals in U.S. history, they weren’t.

Showing a dramatic burst sure to captivate the folks back home, the Americans packed an entire World Cup’s worth of theatrics into a 15-minute span by beating Brazil 5-3 on penalty kicks after a 2-2 tie Sunday night.

Abby Wambach tied it with a magnificent, leaping header in the 122nd minute, and Hope Solo denied the Brazilians — again — in one of the most riveting games in the history of the World Cup, men’s or women’s.

“There is something special about this group. That energy, that vibe,” Solo said. “Even in overtime, you felt something was going to happen.”

The United States advanced to Wednesday’s semifinals against France, which eliminated England on penalty kicks Saturday. And while the Americans will have to win twice more to win the final, they are the only one of the favorites left after two-time defending champ Germany was stunned by Japan on Saturday night.

The U.S. victory came 12 years to the day the Americans last caught their country’s attention in a big way with their penalty-kick shootout victory over China at the Rose Bowl that gave them their second World Cup title.

For Brazil, it is yet another disappointment at a major tournament. And this one is sure to sting more than any others because Marta had it won for the Brazilians, scoring her second goal of the game in the second minute of overtime for the 2-1 lead. But Erika stalled when she went down on a tackle, and the delay added three minutes of stoppage time to the game.

That was all the time Wambach and the Americans needed, after pushing themselves to limit while playing a woman short after Rachel Buehler’s 66th-minute ejection.

Two minutes into stoppage time, Megan Rapinoe blasted a left-footed cross from 30 yards out on the left side that Andreia didn’t come close to getting her hands on. Wambach, one of the best players in the world in the air, made contact and with one furious whip of her head, buried it in the near side of the net from about five yards.

“I took a touch and smoked it,” Rapinoe said. “I don’t think I’ve ever hit a cross with my left foot that well. And then that beast in the air got ahold of it.”

Wambach let out a primal scream and slid into the corner, pumping her fists and quickly mobbed by teammates. No goal had ever been scored that deep into a World Cup game.

“Everything seemed to be on the safe side, but it wasn’t,” Brazil coach Kleiton Lima said. “Unfortunately there was the goal.”

The Americans, shooting first, made their three penalty kicks only to have Cristiane and Marta easily match them. But then it was Daiane’s turn — the same Daiane who’d given the U.S. a 1-0 lead with an own goal in the second minute of the game. She took a hard shot, but Solo stretched out and batted it away. Though the U.S. still had to make two more, the celebration was already starting.

After Rapinoe blistered the net with a blast and Ali Krieger converted hers, the Americans raced onto the field, their joy only matched by that of the pro-American crowd of 25,598. Wambach tackled Solo and U.S. coach Pia Sundhage even broke out her air guitar when AC/DC’s “You Shook Me All Night Long” began to play.

“It is a special moment for me and for this team,” Solo said.

It’s redemption for the rest of the Americans, too, who have been roundly criticized and questioned for their uncharacteristically inconsistent play in recent months. After going more than two years without a loss, they’ve been beaten four times since November.

“It’s like a storybook,” Wambach said.

Brazil has never won a major tournament. It lost to the Americans in the two Olympic gold-medal games, and to Germany in the 2007 World Cup final.

The U.S. has now eliminated Brazil at five of the last seven major tournaments. The lone consolation was that Marta’s goals, the 13th and 14th of her career, tied her with Birgit Prinz atop the all-time World Cup scoring list. The Americans also have won their last five meetings against Brazil.

None, however, was more memorable than this.