The third seed from Spain, Carlos Alcaraz, was beaten by the British number one in the quarter-finals.
In just over three hours, Norrie fought back from a 4-1 deficit in the second set to win.
The world number 11 will play Croatia's Borna Coric in the next round, while the world number one will play Greece in the other semi after their victories.
Norrie said that it was unbelievable.
Alcaraz, ranked fourth in the world, was trying to win his third Masters 1000 title of the year but was only able to convert two of the 13 break points he had against Norrie.
The British man has reached his second Masters 1000 semi- final. I lost a bit of vision while I was up a set and break. He raised his level, didn't give me anything, and I was thinking too much about the finish line.
I wanted to hang tough with him, and I think the only place I had him better was his legs and the physical nature. I was trying to make it difficult for him to finish points by making rallies physical.
Norrie is expected to move up to at least ninth in the world rankings.
The US Open is the grand slam of the year.
American John Isner lost to world number seven Tsitsipas, while Croatia's Coric beat Canada's Felix Auger-Aliassime.
Madison Keys will face Petra Kvitova in the semi-finals of the women's draw after she defeated Elenaryckina.
Two-time Wimbledon champion Kvitova, of the Czech Republic, will play Keys in the semi-finals.
The other semi will feature world number seven Aryna Sabalenka.
They all came through with minimal fuss, with the exception of the case of the world number 35, who won her match against the American.
She has won 25 matches in a row since the beginning of June and has earned more respect from the tennis world.
She said that people forget about her when she does bad and that she did good in the past.
It's like you are a top star again after a couple of wins in a row. It's like you go from a loser to one of the best players.
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