Neither player was at her best after a long wait to come on court but, while 11th seed Bencic, 29, started the steadier, seventh seed Gauff, 22, proved the more adventurous, hitting 35 winners to Bencic's 19 though she also produced 46 unforced errors and nine double faults.
Bencic, a semi-finalist here last year and Olympic champion in 2021, took the first set on her third set point when the erratic Gauff dumped a return into the net.
Gauff, a favourite with the Wimbledon crowd since she reached the fourth round as a bubbly 15-year-old in 2019, found her range in the second set, breaking Bencic's serve twice and dropping her own only once. She produced a drop shot and an exquisite lob to take the match into a third set.
Playing some inspired tennis and finding the corners of the court, Gauff had Bencic on the back foot in the third set.
With the clock ticking down to Wimbledon's 11 pm closure deadline, Gauff set up match point with a smash and won the battle with a big serve that Bencic could not return.
"I'm super happy to be in the quarters finally - I don't know how many tries it's been," Gauff, who will now face fellow American Jessica Pegula, said in a hurried courtside interview.
Bencic, a semi-finalist here last year and Olympic champion in 2021, took the first set on her third set point when the erratic Gauff dumped a return into the net.
Gauff, a favourite with the Wimbledon crowd since she reached the fourth round as a bubbly 15-year-old in 2019, found her range in the second set, breaking Bencic's serve twice and dropping her own only once. She produced a drop shot and an exquisite lob to take the match into a third set.
"I was looking at the clock in the last service game and on that match point I was going for serve and volley as I was wanting to end the point.
"I'm not used to racing for time as we're not used to having to finish by a certain time."
The All England club is in a residential district of southwest London and has a strict curfew.
Gauff said she was hungry for more after finally reaching the quarter-finals.
"Even though it was a tough match I feel this was my best match of the tournament," she said.