24 year old Vondrousova, had not faced a break point until then but when serving for the set at 5-4 she hit two double faults before the four-time Grand Slam champion broke her to love and then rallied to win the tie-break.
Swiatek then powered through the second set, breaking serve three times to reach her 10th semi-final of the year.
Vondrousova, who claimed a shock win at SW19 earlier this summer, was competing in her first WTA 1000 quarter-final in more than three years.
World number seven Gauff will face 43rd-ranked Paolini in the quarter-finals after the Italian’s last-16 opponent Elena Rybakina retired because of injury.
Elsewhere in the last eight,Africa’s Ons Jabeur plays world number two Aryna Sabalenka in a repeat of their Wimbledon semi-final, which Tunisia’s Jabeur fought back to win.