The 22-year-old from Switzerland, who knocked out defending champion and top seed Naomi Osaka in the fourth round, won 7-6 (5) 6-3 against her close friend and frequent hitting partner.
She said: "I think we are professional enough to be friends off the court but do our work on the court and stay focused. I'm sure off the court we are still friends.
"I really like the challenge. Some players are maybe afraid of the big courts but for me it is more motivation. I am really enjoying my tennis."
Croatian Vekic, seeded 23, broke for 5-4 and served for the opening set but was unable to close it out.
She saved a set point at 5-6 and two more in the tie-break, but a long forehand gave Bencic the upper hand.
Bencic, a former Eastbourne winner, had a huge slice of luck serving at 30-30 at the start of the second set when a backhand was flying wide but clipped the net cord twice before landing in.
Vekic saved two break points at 2-2 but Bencic cranked up the pressure and broke her next service game to love before reeling off the final three games and converting her third match point.