Bencic will play Thursday against the winner of a later quarter-final between Canadian 15th seed Bianca Andreescu and Belgian 25th seed Elise Mertens.
One of them will become a first-time Grand Slam finalist in Saturday's championship match at Arthur Ashe Stadium, where first-time Slam winners have been crowned three of the past four years.
"I dreamed of this like a little kid so now that I'm here I'm really enjoying it," Bencic said.
Bencic ensured a return to the world rankings top-10 for the first time since June 2016 with the victory.
She sank as low as 328th after numerous injuries and 2017 left wrist surgery but the 22-year-old Swiss began the year 55th and has climbed from there, thanks in part to her third career WTA title in February at Dubai.
"Through the tough times it helped me to see a little different way the tennis when you can't always play," Bencic said. "For me this perspective changed and I'm enjoying so much the tennis now."
Bencic, who has never faced Andreescu or Mertens, improved to 3-1 against her friend Vekic, who ousted her from this year's French Open.
"We're both professionals enough to be friends off the court and competitive on the court," Bencic said. "I think we did well. I'm sure off the court we are friends still."
Bencic, who enjoyed a third-round walkover, ousted top-ranked defending champion Naomi Osaka in the fourth round, assuring the Japanese star will be replaced atop the rankings on Monday by French Open champion Ashleigh Barty of Australia.
Vekic and Bencic exchanged baseline groundstrokes most of the match. They traded breaks in the ninth and 10th games of the first set and Vekic saved a break point in the 12th game with a forehand volley winner to hold into the tie-breaker.
Bencic went on a 5-1 tie-break run to seize a 6-3 edge and took the set after an hour when Vekic swatted a forehand beyond the baseline.
Vekic netted a backhand to surrender a break and hand Bencic a 4-3 edge in the second set. Bencic held and broke again to close out the match after one hour 41 minutes when Vekic hit a forehand wide, the last of her 28 unforced errors.
- Teenage dreams -
There will be four different women crowned as Grand Slam champions in 2019. It's the third consecutive year for the feat, an Open Era first.
Andreescu, at 19, would be the first teen US Open semi-finalist since Caroline Wozniacki in 2009.
She could become the youngest US Open champion since Russia's Svetlana Kuznetsova, who was a week younger than the suburban Toronto product when she won in 2004.
Andreescu, a champion this year at Indian Wells and Toronto, would be the first teen to win a Slam since Maria Sharapova took the 2006 US Open.
But to have that chance, the Canadian must first dispatch Mertens, who dropped a tournament-low 16 games through four rounds.
Mertens was the only quarter-finalist in her half of the draw to reach a Slam semi-final, having done so at last year's Australian Open.