Anderson's win ended the U.S. presence in the men's draw while Sloane Stephens and Venus Williams have both qualified for the last four in the women's draw.
The 17th-seeded Querrey can blame himself for a meltdown in the first set tiebreak, where he lost five points in a row to allow Anderson into the lead under the lights on Arthur Ashe Stadium.
The 31-year-old Anderson, seeded 28th, will face Spanish 12th seed Pablo Carreno Busta for a spot in Sunday's final.
"This is incredible, at this stage and playing on one of the most famous courts in the world, it feels absolutely fantastic," said Anderson, who became the first South African player to reach the last four of the U.S. Open since tennis turned professional in 1968.
News that makes sense
Your trusted source for staying up-to-date with the world around you. Get free daily news updates and analysis, straight to your inbox.
"Losing the second set was really tough but I played each point at a time and it paid big dividends for me today."
The two big servers could not manage a break in the opening set and Querrey, who was hoping to be the first American male to make it to the last four here since Andy Roddick in 2006, moved 5-2 up in the tiebreaker but Anderson then played more freely and won five points in a row to take the lead.
There was an early exchange of breaks in the second set with Querrey opening a 6-1 lead in the tiebreak before suffering another meltdown and finding himself trailing 8-7.
This time, however, the 17th seed regrouped and equalised on his eighth set point when Anderson returned long, triggering a huge roar from the crowd.
But the stands gradually cleared up as the match was sent deep into the night.
Anderson got a decisive break in the sixth game of the third set as Querrey struggled with his first serve and the South African held twice, finishing it off with a forehand winner down the line to lead two sets to one.
The fourth set was tightly contested again and went into another tiebreak.
Querrey saved a first match point with an ace at 6-5, set up a set point at 7-6, which Anderson staved off by forcing his opponent to slice a backhand into the net.
The second match point was good enough for Anderson, who dominated the rally until Querrey's forehand sailed long.
(Reporting by Julien Pretot; Editing by Sudipto Ganguly)
