Halep was comprehensively beaten 6-4 6-1 by Vandeweghe, who has now won her previous three matches against world number ones.
"It's always something special when you beat a world number one, no matter what stage it is, what venue," Vandeweghe said in quotes published on wtatennis.com.
"To get a victory over a world number one it's a certain pedigree and you definitely hope that you come out at the end with a win. I was fortunate enough to be able to do that."
Vandeweghe will now face Caroline Garcia in the last four after she came from a set down to beat third seed Elina Svitolina 6-7 (4/7) 6-4 6-2.
Anett Kontaveit overcame Anastasia Pavlyuchenkova in three sets and Karolina Pliskova also went the distance in beating Jelena Ostapenko.
At the Istanbul Cup, world number two Caroline Wozniacki was forced to retire due to a left abdominal injury, while Svetlana Kuznetsova was also eliminated at the quarter-final stage.
Dane Wozniacki, the top seed in Turkey, won the first set against the unseeded Pauline Parmentier, who then took the second set before Wozniacki withdrew through injury.
"I saw she had some problems in the second set with the serve," Parmentier said.
"She was not serving full speed. It's tough when you see your opponent not doing well."
Second seed Kuznetsova lost a third-set decider to Slovenian Polona Hercog, who progressed to her first WTA semi-final in over two years courtesy of a 6-1 1-6 7-5 success.
Hercog will meet Maria Sakkari in the semi-finals following the Greek's straight-sets win over qualifier Arantxa Rus.
Parmentier will now face Irina Camelia-Begu, the only remaining seed, in the last four after the Romanian came from a set down to win 3-6 7-6 (7/1) 6-1 against Donna Vekic.
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