Thompson started at setter in the straight-set victory over the Dominicans after team captain Lindsey Berg injured her left ankle in the previous match.
"Courtney's a quarterback," Hooker said. "She went out there and took control of the game. She knows our rhythm and it was great to have her on the court."
The Americans remained undefeated in six matches at the London Olympics with the 25-14, 25-21, 25-22 sweep of the Dominicans.
The top-ranked U.S. will play South Korea in Thursday's semifinals. Japan and Brazil face off in the other match.
Berg's status is uncertain. The Americans are not revealing too much about the injury because it would give their opponents a competitive advantage.
"It's day to day," coach Hugh McCutcheon said. "I thought Courtney did a great job. I don't think we missed too many beats tonight."
Hooker scored 19 points for the U.S., which has never won a gold in indoor volleyball. The team took silver at the 2008 Beijing Olympics after falling in the final to Brazil.
The Americans jumped to a 12-3 lead and rolled through the first set, winning it on Foluke Akinradewo's block.
While clearly overmatched, the Dominican Republic took an 8-6 lead in the second set. The U.S. regained the lead when Hooker's spike made it 11-10. The Dominicans denied the U.S. set point three times, but Jordan Larson spiked to win it.
The U.S. took an 18-12 lead on Christa Harmotto's spike, but the Dominicans closed to 19-18 before the Americans won it on Bethania de la Cruz de Pena's misplayed kill.
Now playing under McCutcheon, who led the U.S. men to the gold in Beijing, the United States is among the favorites in the tournament.
The Americans took over the top spot in the world rankings at the end of 2011, ending Brazil's four-year run. The women qualified for the Olympics in November with a silver-medal finish in the FIVB World Cup.
The U.S. beat No. 2 Brazil 3-1 in a preliminary-round rematch of the Beijing final last week.
The 11th-ranked Dominican Republic finished 2-3 in pool play, beating Algeria and Britain in straight sets. The Dominicans' only other Olympic appearance was at the Athens Games, where they upset the United States 3-2 but finished 11th.
The Americans have defeated the Dominican Republic three times this year, most recently in a four-set semifinal at the Pan American Cup.
Prisilla Altagracia Rivera Brens finished with 15 points for the Dominican Republic.
It was unclear whether the U.S. held out Berg, a three-time Olympian, to give her extra rest before the semifinals, or whether it was because of the ankle injury. She dressed and warmed up with the team with no limp.
The 27-year-old Thompson helped the Americans to a bronze medal at the 2011 Pan American Games, including the four-set win over the Dominicans.
Often during timeouts, Berg pulled Thompson aside to confer.
"It was great, I had fun and I was thankful," said Thompson, who is playing in her first Olympics. "My teammates are amazing."
Berg would not comment about the injury after the match.
"I think any 12 can be on the court and we can be successful," she said.
The entire Dominican team attended the post-match news conference, many of them crying, to honor captain Milagros Cabral del la Cruz, who is retiring.
"This is a very young team who played against one of the best teams in the world at a very high level," Cabral de la Cruz said.
Earlier Tuesday, Brazil edged Russia 3-2, and Japan beat No. 3 China 3-2.
The 15th-ranked South Koreans rallied after dropping the first set against No. 4 Italy, winning 3-1 in the final quarterfinal of the day.
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