The Polish player will face Spain’s Carlos Alcaraz in the semi-finals
The Pole endured the world number two’s best shots early to grind out a tough opening set 7-6 (7) before disarming Medvedev with some spectacular net play to take the second 6-3.
In comments carried by the ATP Tour site, the world number 10 said after the match that the humid conditions helped get him over the line.
“I have spent a lot of time in Florida, so I am used to the humidity,” he said.
“I think the conditions were in my favour today, so I tried to use them.
“I think the return was crucial. I was able to make a lot of returns and get some free points on my serve, because having rallies with Daniil is fun, but they get long. It is good I was able to get some free points.”
Hurkacz will face Spain’s Carlos Alcaraz in the semi-finals after the teenager overcame Serbia’s Miomir Kecmanovic 6-7 (5) 6-3 7-6 (5).
Although Bencic raced out of the gates with some superb returns, Osaka found her serve in the second as she sent down 18 aces en route to a 4-6 6-3 6-4 victory.
She will meet soon-to-be world number one Iga Swiatek in the final after the Pole defeated Jessica Pegula of the US 6-2 7-6 in their semi-final.