Orozco's 80th-minute goal was complemented by a pair of fine saves by US goalkeeper Tim Howard in the last 10 minutes to secure victory at Mexico City's intimidating Azteca stadium.
US coaches hugged on the sideline on the final whistle while American players dropped to their knees in celebration.
"It was a fantastic team performance," US manager Jurgen Klinsmann said.
"Getting that win here, it's quite enjoyable. I think it's huge for all American fans and for the team. We were well aware we'd never won at Azteca stadium."
Facing a dangerous attack led by striker Javier Hernandez, the US surrendered possession for most of the match and appeared content to escape with a draw.
But the game turned when Brek Shea made a surprise run on the left wing and played a ball that was back-heeled by Terrance Boyd and knocked in goal by Orozco.
Mexico entered the match with a 23-0-1 home record against the Americans whose previous best result on Mexican soil was 0-0 in 1998.
The US's triumph was built on a watertight defence that nullified the hosts' relentless attacking.
Mexico finished with 19 shots on goal to the visitors' seven, and enjoyed the lion's share of possession.
Hernandez came close to breaking US hearts in the 88th minute with a spectacular header, but Howard saved it to bring the Mexicans back to earth after their Olympic team's gold medal triumph at the London Games last week.